user17410: While I do have a personal and intimate belief in reincarnation, I really struggle with the Buddhist belief that we have innumerably many past lives.
A monk I listened to explained that all the beings we meet everyday were once our parents in a specific past life, as well as our enemies.
Honestly, i find this perspective quite hard to believe.
It seems to me that if had so many past lives, we should have reached wisdom and Buddhahood at some point already.
How (according to Buddhism) is it possible to have such an accumulation of experience, without reaching some degree of wisdom or enlightenment?

user13709: Infinities can be one-way. Consider the negative numbers.
What the Buddha said is that there is no "first point".

SN15.19:1.3: No first point is found of sentient beings roaming and transmigrating, hindered by ignorance and fettered by craving.

And with the ending of ignorance, one does not return.

AN2.36:4.7: When they pass away from there, they are non-returners, not coming back to this state of existence.

Non-returners (anāgāmi) and arahants do not return. They have left behind the negative infinity of suffering.

user18: It is said to have taken the last Buddha more than four uncountable time periods from the time he received confirmation from an earlier Buddha to become a Buddha. That's already a fairly unfathomably long period of time, and he was of course a very special, rare exception to the norm. Even among Buddhas his journey is said by the commentaries to have been relatively short (it can take up to more than eighty uncountable periods, iirc). 
Becoming enlightened isn't just something one falls into accidentally. It is beyond all forms of worldly wisdom and requires exceedingly special circumstances to occur. Even worldly insight into reality that might help us reach enlightenment comes and goes and is replaced by ignorance and delusion, so just because we may have at times gained worldly wisdom in the past does not mean we will eventually become enlightened as a result.

user471: From SN 15.3:

“...  What do you think? Which is more: the flow of tears you’ve shed
  while roaming and transmigrating for such a very long time—weeping and
  wailing from being united with the unloved and separated from the
  loved—or the water in the four oceans?”
“As we understand the Buddha’s teaching, the flow of tears we’ve shed
  while roaming and transmigrating is more than the water in the four
  oceans.”
“Good, good, mendicants! It’s good that you understand my teaching
  like this. The flow of tears you’ve shed while roaming and
  transmigrating is indeed more than the water in the four oceans. For a
  long time you’ve undergone the death of a mother … father … brother …
  sister … son … daughter … loss of relatives … loss of wealth … or loss
  through illness. From being united with the unloved and separated from
  the loved, the flow of tears you’ve shed while roaming and
  transmigrating is indeed more than the water in the four oceans.
Why is that? Transmigration has no known beginning. … This is quite
  enough for you to become disillusioned, dispassionate, and freed
  regarding all conditions.”

It's true that according to the sutta above, you have been reborn innumerable times, with no known beginning to suffering.
But usually, people take this to mean that the very same consciousness that is reading this sentence has been reborn many times and experienced all that has been experienced through this body since birth.
However, according to MN 38, this is not the case:

The Blessed One then asked him: “Sāti, is it true that the following
  pernicious view has arisen in you: ‘As I understand the Dhamma taught
  by the Blessed One, it is this same consciousness that runs and
  wanders through the round of rebirths, not another’?”
“Exactly so, venerable sir. As I understand the Dhamma taught by the
  Blessed One, it is this same consciousness that runs and wanders
  through the round of rebirths, not another.”
“What is that consciousness, Sāti?”
“Venerable sir, it is that which speaks and feels and experiences here
  and there the result of good and bad actions.”
“Misguided man, to whom have you ever known me to teach the Dhamma in
  that way? Misguided man, have I not stated in many ways consciousness
  to be dependently arisen, since without a condition there is no
  origination of consciousness? But you, misguided man, have
  misrepresented us by your wrong grasp and injured yourself and stored
  up much demerit; for this will lead to your harm and suffering for a
  long time.”

Consciousness is always changing and it is dependently originated. However, it may appear to be the same just as the water stream of a river appears to be the same, when in fact, the droplets of water in it are constantly changing. Please see this answer.
So, yes - you have been reborn innumerable number of times and have shed so much tears mourning the deaths of innumerable loved ones. But who are "you"? You are not the same consciousness that moves through the births.
"You" refers to the self i.e. the thought or idea of the self. It's not a soul. It's simply the thought or idea or view of the self in the mind. It's not the same common self for every person. It's also not the same thought that exists at all moments in time - it's dependently originated.
When there is the thought of the self, there is the objectification and classification (papanca) of everything else that is perceived as non-self, relative to its relationship to the self. This causes the arising of craving and suffering. This is elaborated in this answer.
So, there is definitely rebirth - not the rebirth of a permanent consciousness or soul, but rather the rebirth of the mental idea of the self, and together with it, the rebirth of suffering.
So, the Buddha looks at YOU - yes, that's right - YOU - the SELF - and tells you that you have been reborn innumerable number of times, with no known beginning. You (the SELF) have also shed oceans of tears, mourning the deaths of innumerable loved ones of your's. This to me, is definitely proof that there is rebirth in Buddhism, just not the rebirth that you may have imagined.
And who says that you have not reached some level of wisdom? The very fact that you are writing this question or reading this answer, shows that you have at least enough wisdom to consider that you should seek the end of suffering.

user13375: This question was beguiling because on the surface it appears to be about two different subjects that are not necessarily Buddhist: various doctrines of rebirth/reincarnation and the mathematical treatment of infinity. However, after clarification and looking at Ruben's excellent answer I see that that my actual answer from a Buddhist perspective is only tangentially related to these because the core of the question actually involves a subtle assumption that I think is false.
First, let's reiterate and emphasize what Ruben laid out. The Buddhist doctrine of rebirth when properly understood is not one of reincarnation. The Buddha specifically and emphatically refuted the idea of an atman that transmigrates from life to life. Moreover, and even more importantly he specifically refuted that an atman exists at all! That is, even in this very life... there is no real self that inhabits the body and carries over from moment to moment. So let's just set aside rebirth for now and concentrate on this very life and what happens from moment to moment.
In this very life, sentient beings go around from moment to moment believing and acting as if there really is a soul or atman inhabiting their body and mind. That there is some unchanging entity that exists continuously and that is the proper object that we refer to as "I" and is the owner of "Mine" and is distinct from all other things that inhabit the world whether they be other sentient beings or inanimate objects. Isn't this the case? Don't we all feel very deeply and believe and act as if this was the case?
Now, if you look at the question and take out the rebirth and talk of infinity the question might very well be formulated as, "How is it possible to have such an accumulation of experience from moment to moment in this very life without reaching some degree of wisdom or enlightenment?" That is, as a sentient being grows older in this very life how is it not possible that we'll end up wiser and closer to enlightenment near the end of our life compared to the beginning of our life?
And that's the subtle assumption I was talking about earlier... Is it actually true that sentient beings inexorably end up with more wisdom and closer to enlightenment at the end of life as compared to the beginning? The real question is whether this subtle assumption is true. Do we inevitably get more wise with age? Do we become more enlightened? Here's how I would definitively formulate the question/assumption at the heart of this post:

Is wisdom and enlightened activity subject to some universal law that
  says it inevitably increases on longer times scales even if subject to
  some ups and downs on shorter time scales?

I think with just a little bit of reflection and examination of experience we can see that this assumption is just not true and the answer to the question is an emphatic no. Lots and lots of sentient beings are born, grow older, enter adult life, grow sick and die and there is no net gain in wisdom or enlightened activity. Many many beings end up less wise than when they began. They might have been happy children who made wrong life choices or decisions and ended up in miserable adult lives. Or they might have been met with traumatic brain injury due to karma and their intelligence and wisdom could have dropped precipitously.
It is just not true that in this very life with the accumulation of experiences that wisdom or enlightened activity necessarily increase.
Which brings us back to rebirth...
What many fail to understand is that the Buddha taught that the process of rebirth for transmigrating beings is exactly the same thing that happens from moment to moment in this very life. To the EXACT SAME EXTENT AND MANNER that sentient beings exist and go from moment to moment in this very life... they go from life to life. There is nothing whatsoever different in the manner that beings transmigrate in this very life to the manner in which they transmigrate from life to life. To think there is a difference is to not understand the Buddha's most subtle teachings.
Atman does not exist even in the slightest. The way sentient beings exist and move from moment to moment in this very life is like an illusion and subject to dependent arising. There is no universal law that wisdom accumulates on longer time scales as compared to smaller time scales. The way beings exist and move from life to life is like an illusion and subject to dependent arising. The belief that atman exists and the ignorance that it does not exist is what propels beings from moment to moment and from life to life in samsara.
Sincerely hope this answer is helpful and wish you well-being Ian.

user43: As I understand, the Buddha meant that we collectively had gone through an uncountable number of existances, in all shapes and forms. From this mass of existances, some were able to attain Enlightenment (thereby arising as the Buddhas and their students) and some weren't.
Wisdom and knowledge is definitely accumulated, but not everyone can take advantage of it. The reason for that is karma, which is the law of self-reinforcing tendencies. If someone steps on the wrong path, they create circumstances that lead further and further away from those situations where the wisdom is known to manifest -perhaps delaying or making it harder to get out of that loop.

17410: <p>While I do have a personal and intimate belief in reincarnation, I really struggle with the Buddhist belief that we have innumerably many past lives.</p>

<p>A monk I listened to explained that all the beings we meet everyday were once our parents in a specific past life, as well as our enemies.
Honestly, i find this perspective quite hard to believe.
It seems to me that if had so many past lives, we should have reached wisdom and Buddhahood at some point already.</p>

<p>How (according to Buddhism) is it possible to have such an accumulation of experience, without reaching some degree of wisdom or enlightenment?</p>
 13709: <p>Infinities can be one-way. Consider the negative numbers.</p>

<p>What the Buddha said is that there is no "first point".</p>

<blockquote>
  <p><a href="https://suttacentral.net/sn15.19/en/sujato#sn15.19:1.3" rel="nofollow noreferrer">SN15.19:1.3</a>: No first point is found of sentient beings roaming and transmigrating, hindered by ignorance and fettered by craving.</p>
</blockquote>

<p>And with the ending of ignorance, one does not return.</p>

<blockquote>
  <p><a href="https://suttacentral.net/an2.32-41/en/sujato#an2.36:4.7" rel="nofollow noreferrer">AN2.36:4.7</a>: When they pass away from there, they are non-returners, not coming back to this state of existence.</p>
</blockquote>

<p>Non-returners (anāgāmi) and arahants do not return. They have left behind the negative infinity of suffering.</p>
 18: <p>It is said to have taken the last Buddha more than four uncountable time periods from the time he received confirmation from an earlier Buddha to become a Buddha. That's already a fairly unfathomably long period of time, and he was of course a very special, rare exception to the norm. Even among Buddhas his journey is said by the commentaries to have been relatively short (it can take up to more than eighty uncountable periods, iirc). </p>

<p>Becoming enlightened isn't just something one falls into accidentally. It is beyond all forms of worldly wisdom and requires exceedingly special circumstances to occur. Even worldly insight into reality that might help us reach enlightenment comes and goes and is replaced by ignorance and delusion, so just because we may have at times gained worldly wisdom in the past does not mean we will eventually become enlightened as a result.</p>
 471: <p>From <a href="https://suttacentral.net/sn15.3/en/sujato" rel="nofollow noreferrer">SN 15.3</a>:</p>

<blockquote>
  <p>“...  What do you think? Which is more: the flow of tears you’ve shed
  while roaming and transmigrating for such a very long time—weeping and
  wailing from being united with the unloved and separated from the
  loved—or the water in the four oceans?”</p>
  
  <p>“As we understand the Buddha’s teaching, the flow of tears we’ve shed
  while roaming and transmigrating is more than the water in the four
  oceans.”</p>
  
  <p>“Good, good, mendicants! It’s good that you understand my teaching
  like this. The flow of tears you’ve shed while roaming and
  transmigrating is indeed more than the water in the four oceans. For a
  long time you’ve undergone the death of a mother … father … brother …
  sister … son … daughter … loss of relatives … loss of wealth … or loss
  through illness. From being united with the unloved and separated from
  the loved, the flow of tears you’ve shed while roaming and
  transmigrating is indeed more than the water in the four oceans.</p>
  
  <p>Why is that? Transmigration has no known beginning. … This is quite
  enough for you to become disillusioned, dispassionate, and freed
  regarding all conditions.”</p>
</blockquote>

<p>It's true that according to the sutta above, you have been reborn innumerable times, with no known beginning to suffering.</p>

<p>But usually, people take this to mean that the very same consciousness that is reading this sentence has been reborn many times and experienced all that has been experienced through this body since birth.</p>

<p>However, according to <a href="https://suttacentral.net/mn38/en/bodhi" rel="nofollow noreferrer">MN 38</a>, this is not the case:</p>

<blockquote>
  <p>The Blessed One then asked him: “Sāti, is it true that the following
  pernicious view has arisen in you: ‘As I understand the Dhamma taught
  by the Blessed One, it is this same consciousness that runs and
  wanders through the round of rebirths, not another’?”</p>
  
  <p>“Exactly so, venerable sir. As I understand the Dhamma taught by the
  Blessed One, it is this same consciousness that runs and wanders
  through the round of rebirths, not another.”</p>
  
  <p>“What is that consciousness, Sāti?”</p>
  
  <p>“Venerable sir, it is that which speaks and feels and experiences here
  and there the result of good and bad actions.”</p>
  
  <p>“Misguided man, to whom have you ever known me to teach the Dhamma in
  that way? Misguided man, have I not stated in many ways consciousness
  to be dependently arisen, since without a condition there is no
  origination of consciousness? But you, misguided man, have
  misrepresented us by your wrong grasp and injured yourself and stored
  up much demerit; for this will lead to your harm and suffering for a
  long time.”</p>
</blockquote>

<p>Consciousness is always changing and it is dependently originated. However, it may appear to be the same just as the water stream of a river appears to be the same, when in fact, the droplets of water in it are constantly changing. Please see <a href="https://buddhism.stackexchange.com/a/30700/471">this answer</a>.</p>

<p>So, yes - you have been reborn innumerable number of times and have shed so much tears mourning the deaths of innumerable loved ones. But who are "you"? You are not the same consciousness that moves through the births.</p>

<p>"You" refers to the self i.e. the thought or idea of the self. It's not a soul. It's simply the thought or idea or view of the self in the mind. It's not the same common self for every person. It's also not the same thought that exists at all moments in time - it's dependently originated.</p>

<p>When there is the thought of the self, there is the objectification and classification (<em>papanca</em>) of everything else that is perceived as non-self, relative to its relationship to the self. This causes the arising of craving and suffering. This is elaborated in <a href="https://buddhism.stackexchange.com/a/31935/471">this answer</a>.</p>

<p>So, there is definitely rebirth - not the rebirth of a permanent consciousness or soul, but rather the rebirth of the mental idea of the self, and together with it, the rebirth of suffering.</p>

<p>So, the Buddha looks at YOU - yes, that's right - YOU - the SELF - and tells you that you have been reborn innumerable number of times, with no known beginning. You (the SELF) have also shed oceans of tears, mourning the deaths of innumerable loved ones of your's. This to me, is definitely proof that there is rebirth in Buddhism, just not the rebirth that you may have imagined.</p>

<p>And who says that you have not reached some level of wisdom? The very fact that you are writing this question or reading this answer, shows that you have at least enough wisdom to consider that you should seek the end of suffering.</p>
 13375: <p>This question was beguiling because on the surface it appears to be about two different subjects that are not necessarily Buddhist: various <strong>doctrines of rebirth/reincarnation</strong> and the <strong>mathematical treatment of infinity.</strong> However, after clarification and looking at <a href="https://buddhism.stackexchange.com/a/36839/13375">Ruben's excellent answer</a> I see that that my actual answer from a Buddhist perspective is only tangentially related to these because <strong>the core of the question actually involves a subtle assumption</strong> that I think is false.</p>

<p>First, let's reiterate and emphasize what Ruben laid out. The Buddhist doctrine of rebirth when properly understood is not one of reincarnation. The Buddha specifically and emphatically refuted the idea of an atman that transmigrates from life to life. Moreover, and even more importantly he specifically refuted that an atman exists at all! That is, even in this very life... there is no real self that inhabits the body and carries over from moment to moment. So let's just set aside rebirth for now and concentrate on this very life and what happens from moment to moment.</p>

<p>In this very life, sentient beings go around from moment to moment believing and acting as if there really is a soul or atman inhabiting their body and mind. That there is some unchanging entity that exists continuously and that is the proper object that we refer to as <strong>"I"</strong> and is the owner of <strong>"Mine"</strong> and is distinct from all other things that inhabit the world whether they be other sentient beings or inanimate objects. Isn't this the case? Don't we all feel very deeply and believe and act as if this was the case?</p>

<p>Now, if you look at the question and take out the rebirth and talk of infinity the question might very well be formulated as, <strong>"How is it possible to have such an accumulation of experience from moment to moment in this very life without reaching some degree of wisdom or enlightenment?"</strong> That is, as a sentient being grows older <strong>in this very life</strong> how is it not possible that <strong>we'll end up wiser and closer to enlightenment near the end of our life compared to the beginning of our life?</strong></p>

<p>And that's the subtle assumption I was talking about earlier... Is it actually true that sentient beings inexorably end up with more wisdom and closer to enlightenment at the end of life as compared to the beginning? The real question is whether this subtle assumption is true. Do we inevitably get more wise with age? Do we become more enlightened? Here's how I would definitively formulate the question/assumption at the heart of this post:</p>

<blockquote>
  <p><strong>Is wisdom and enlightened activity subject to some universal law that
  says it inevitably increases on longer times scales even if subject to
  some ups and downs on shorter time scales?</strong></p>
</blockquote>

<p>I think with just a little bit of reflection and examination of experience we can see that this assumption is just not true and the answer to the question is an emphatic no. Lots and lots of sentient beings are born, grow older, enter adult life, grow sick and die and there is no net gain in wisdom or enlightened activity. Many many beings end up less wise than when they began. They might have been happy children who made wrong life choices or decisions and ended up in miserable adult lives. Or they might have been met with traumatic brain injury due to karma and their intelligence and wisdom could have dropped precipitously.</p>

<p>It is just not true that in this very life with the accumulation of experiences that wisdom or enlightened activity necessarily increase.</p>

<p>Which brings us back to rebirth...</p>

<p>What many fail to understand is that the Buddha taught that the process of rebirth for transmigrating beings is exactly the same thing that happens from moment to moment in this very life. To the <strong>EXACT SAME EXTENT AND MANNER</strong> that sentient beings exist and go from moment to moment in this very life... they go from life to life. There is nothing whatsoever different in the manner that beings transmigrate in this very life to the manner in which they transmigrate from life to life. To think there is a difference is to not understand the Buddha's most subtle teachings.</p>

<p>Atman does not exist even in the slightest. The way sentient beings exist and move from moment to moment in this very life is like an illusion and subject to dependent arising. There is no universal law that wisdom accumulates on longer time scales as compared to smaller time scales. The way beings exist and move from life to life is like an illusion and subject to dependent arising. The belief that atman exists and the ignorance that it does not exist is what propels beings from moment to moment and from life to life in samsara.</p>

<p>Sincerely hope this answer is helpful and wish you well-being Ian.</p>
 43: <p>As I understand, the Buddha meant that we <em>collectively</em> had gone through an uncountable number of existances, in all shapes and forms. From this mass of existances, some were able to attain Enlightenment (thereby arising as the Buddhas and their students) and some weren't.</p>

<p>Wisdom and knowledge is definitely accumulated, but not everyone can take advantage of it. The reason for that is karma, which is the law of self-reinforcing tendencies. If someone steps on the wrong path, they create circumstances that lead further and further away from those situations where the wisdom is known to manifest -perhaps delaying or making it harder to get out of that loop.</p>
