user11541: According to dependent origination I took birth (i.e I came into existence) due to ignorance. 
My questions are :how many times it has happened that someone has taken birth into this samsara? 
Since when birth of ignorance has been happening?
If there are no answers in the scriptures to the above questions then what is your personal opinion on the issues mentioned above?

user12365: Very importand question! Why? Because it's required to see the meaninglessness of being a recidivist to actually change ones ways. Such a question and remembering the answer is not one to put aside but to repeat till it bring fruits, till one sees the recidivist clearly! (speech can cause so much... this or that and best.) The question and answer around it teaches not only the terrible size of suffering, but also gives when clear the right amount of effort to stop this foolish recidivist-hood.
Anamatagga-samyutta — The unimaginable beginnings of samsara

"From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries/the death of a father... the death of a brother... the death of a sister... the death of a son... the death of a daughter... loss with regard to relatives... loss with regard to wealth... loss with regard to disease/experienced stress, experienced pain, experienced loss, swelling the cemeteries/...blood you have shed when, being water buffaloes, you had your water buffalo-heads cut off... when, being rams, you had your ram-heads cut off... when, being goats, you had your goat-heads cut off... when, being deer, you had your deer-heads cut off... when, being chickens, you had your chicken-heads cut off... when, being pigs, you had your pig-heads cut off/...arrested as thieves plundering villages, you had your heads cut off... when, arrested as highway thieves, you had your heads cut off... when, arrested as adulterers, you had your heads cut off/... — enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released."

[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]

user9861: The traditional answer is that you have been reborn countless times. Ordinarily, I do not offer my personal opinion on this website. But only because you asked, based upon my meditation on the evolution of consciousness, this traditional answer is accurate because a sentient being is born with the intelligence of a tiny biological organism. Only by being reborn into a symbiotic relationship with progressively more complex organisms does a being acquire human intelligence.

user-1: 
My questions are :how many times it has happened that someone has taken birth into this samsara?
Since when birth of ignorance has been happening?

Buddhism does not answer such questions, since they do not lead to Nibbana. The Buddha said himself, that he teaches only the cessation of suffering.
The questions fall under the Four Imponderables (Acinteyyas), namely the third and forth Acinteyya:

The four imponderables are identified in the Acintita Sutta, Anguttara Nikaya 4.77, as follows:
The Buddha-range of the Buddhas [i.e., the range of powers a Buddha develops as a result of becoming a Buddha];
The jhana-range of one absorbed in jhana [i.e., the range of powers that one may obtain while absorbed in jhana];
The [precise working out of the] results of kamma;
Speculation about [the origin, etc., of] the cosmos is an imponderable that is not to be speculated about.

It's not that these questions are not interesting, they certainly are, but they will not lead one to Nibbana. They will only distract and agitate the mind, leaving it unfit for practice.
If you really want to know the answer to these questions, not only intellectually but experientially, then you have to become a fully enlightened Buddha (Sammāsambuddha).

user471: According to the Assu Sutta:

There the Blessed One said: "From an inconstruable beginning comes
  transmigration. A beginning point is not evident, though beings
  hindered by ignorance and fettered by craving are transmigrating &
  wandering on. What do you think, monks: Which is greater, the tears
  you have shed while transmigrating & wandering this long, long time —
  crying & weeping from being joined with what is displeasing, being
  separated from what is pleasing — or the water in the four great
  oceans?"
"As we understand the Dhamma taught to us by the Blessed One, this is
  the greater: the tears we have shed while transmigrating & wandering
  this long, long time — crying & weeping from being joined with what is
  displeasing, being separated from what is pleasing — not the water in
  the four great oceans."
"Excellent, monks. Excellent. It is excellent that you thus understand
  the Dhamma taught by me.

According to this article, the total amount of tears a person sheds on average in a lifetime is estimated to be 64 litres.
According to this USGS estimation and this article, there is an estimated 1.332 billion cubic kilometers of water in the Earth's oceans. That's 1.332 x 1021 litres.
If I divide that by 64, I get 2.08125 x 1019 lifetimes.
According to the short scale, that's 20 quintillion (20 x 1018).
Hence, according to the Buddha, every person has been reborn at least 20 quintillion times.
However, since the Buddha says that "a beginning point is not evident", I guess that it should be far greater than 20 quintillion times. Maybe a googleplex amount of times.

11541: <p>According to dependent origination I took birth (i.e I came into existence) due to ignorance. </p>

<p>My questions are :how many times it has happened that someone has taken birth into this samsara? </p>

<p>Since when birth of ignorance has been happening?</p>

<p>If there are no answers in the scriptures to the above questions then what is your personal opinion on the issues mentioned above?</p>
 12365: <p>Very importand question! Why? Because it's required to see the meaninglessness of being a recidivist to actually change ones ways. Such a question and remembering the answer is not one to put aside but to repeat till it bring fruits, till one sees the recidivist clearly! (speech can cause so much... this or that and best.) The question and answer around it teaches not only the terrible size of suffering, but also gives when clear the right amount of effort to stop this foolish recidivist-hood.</p>

<p><a href="http://zugangzureinsicht.org/html/tipitaka/sn/index_en.html#sn15" rel="nofollow noreferrer">Anamatagga-samyutta — The unimaginable beginnings of samsara</a></p>

<blockquote>
  <p>"<strong>From an inconstruable beginning comes transmigration. A beginning point is not evident</strong>, though beings hindered by ignorance and fettered by craving are transmigrating &amp; wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries/the death of a father... the death of a brother... the death of a sister... the death of a son... the death of a daughter... loss with regard to relatives... loss with regard to wealth... loss with regard to disease/experienced stress, experienced pain, experienced loss, swelling the cemeteries/...blood you have shed when, being water buffaloes, you had your water buffalo-heads cut off... when, being rams, you had your ram-heads cut off... when, being goats, you had your goat-heads cut off... when, being deer, you had your deer-heads cut off... when, being chickens, you had your chicken-heads cut off... when, being pigs, you had your pig-heads cut off/...arrested as thieves plundering villages, you had your heads cut off... when, arrested as highway thieves, you had your heads cut off... when, arrested as adulterers, you had your heads cut off/... — enough to become disenchanted with all fabricated things, enough to become dispassionate, <strong>enough to be released.</strong>"</p>
</blockquote>

<p><em>[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]</em></p>
 9861: <p>The traditional answer is that you have been reborn countless times. Ordinarily, I do not offer my personal opinion on this website. But <em>only</em> because you asked, based upon my meditation on the evolution of consciousness, this traditional answer is accurate because a sentient being is born with the intelligence of a tiny biological organism. Only by being reborn into a symbiotic relationship with progressively more complex organisms does a being acquire human intelligence.</p>
 -1: <blockquote>
<p>My questions are :how many times it has happened that someone has taken birth into this samsara?</p>
<p>Since when birth of ignorance has been happening?</p>
</blockquote>
<p>Buddhism does not answer such questions, since they do not lead to Nibbana. The Buddha said himself, that he teaches only the cessation of suffering.</p>
<p>The questions fall under the <a href="https://en.wikipedia.org/wiki/Acinteyya#Four_imponderables" rel="nofollow noreferrer"><em>Four Imponderables (Acinteyyas)</em></a>, namely the third and forth Acinteyya:</p>
<blockquote>
<p>The four imponderables are identified in the Acintita Sutta, Anguttara Nikaya 4.77, as follows:</p>
<p>The Buddha-range of the Buddhas [i.e., the range of powers a Buddha develops as a result of becoming a Buddha];</p>
<p>The jhana-range of one absorbed in jhana [i.e., the range of powers that one may obtain while absorbed in jhana];</p>
<p>The [precise working out of the] results of kamma;</p>
<p>Speculation about [the origin, etc., of] the cosmos is an imponderable that is not to be speculated about.</p>
</blockquote>
<p>It's not that these questions are not interesting, they certainly are, but they will not lead one to Nibbana. They will only distract and agitate the mind, leaving it unfit for practice.</p>
<p>If you really want to know the answer to these questions, not only intellectually but experientially, then you have to become a fully enlightened Buddha (Sammāsambuddha).</p>
 471: <p>According to the <a href="https://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html" rel="nofollow noreferrer">Assu Sutta</a>:</p>

<blockquote>
  <p>There the Blessed One said: "From an inconstruable beginning comes
  transmigration. A beginning point is not evident, though beings
  hindered by ignorance and fettered by craving are transmigrating &amp;
  wandering on. What do you think, monks: <strong>Which is greater, the tears
  you have shed while transmigrating &amp; wandering this long, long time —
  crying &amp; weeping from being joined with what is displeasing, being
  separated from what is pleasing — or the water in the four great
  oceans?</strong>"</p>
  
  <p>"As we understand the Dhamma taught to us by the Blessed One, this is
  the greater: <strong>the tears we have shed while transmigrating &amp; wandering
  this long, long time — crying &amp; weeping from being joined with what is
  displeasing, being separated from what is pleasing — not the water in
  the four great oceans.</strong>"</p>
  
  <p>"Excellent, monks. Excellent. It is excellent that you thus understand
  the Dhamma taught by me.</p>
</blockquote>

<p>According to <a href="http://www.lostateminor.com/2014/12/20/chair-filled-many-litres-tears-youll-cry-lifetime/" rel="nofollow noreferrer">this article</a>, the total amount of tears a person sheds on average in a lifetime is estimated to be 64 litres.</p>

<p>According to <a href="https://water.usgs.gov/edu/earthhowmuch.html" rel="nofollow noreferrer">this USGS estimation</a> and <a href="https://www.livescience.com/29673-how-much-water-on-earth.html" rel="nofollow noreferrer">this article</a>, there is an estimated 1.332 billion cubic kilometers of water in the Earth's oceans. That's 1.332 x 10<sup>21</sup> litres.</p>

<p>If I divide that by 64, I get 2.08125 x 10<sup>19</sup> lifetimes.</p>

<p>According to the <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers" rel="nofollow noreferrer">short scale</a>, that's 20 quintillion (20 x 10<sup>18</sup>).</p>

<p>Hence, according to the Buddha, every person has been reborn <strong>at least</strong> 20 quintillion times.</p>

<p>However, since the Buddha says that "a beginning point is not evident", I guess that it should be far greater than 20 quintillion times. Maybe a <a href="https://en.wikipedia.org/wiki/Googolplex" rel="nofollow noreferrer">googleplex</a> amount of times.</p>
