user8157: Here is the Pali from the Nakhasikhā Sutta, which is about the "breakthrough" of three fetters, for a disciple of the noble ones who is consummate in view, an individual who has broken through [to stream-entry]:

Evaṃ me sutaṃ—​   ekaṃ samayaṃ bhagavā sāvatthiyaṃ viharati jetavane
  anātha­piṇḍi­kassa ārāme. Atha kho bhagavā parittaṃ nakhasikhāyaṃ
  paṃsuṃ āropetvā bhikkhū āmantesi: “taṃ kiṃ maññatha, bhikkhave,
  katamaṃ nu kho bahutaraṃ, yo vāyaṃ mayā paritto nakhasikhāyaṃ paṃsu
  āropito, ayaṃ vā mahāpathavī”ti?
“Etadeva, bhante, bahutaraṃ, yadidaṃ mahāpathavī. Appamattako
  bhagavatā paritto nakhasikhāyaṃ paṃsu āropito. Neva satimaṃ kalaṃ
  upeti na sahassimaṃ kalaṃ upeti na satasahassimaṃ kalaṃ upeti
  mahāpathaviṃ upanidhāya bhagavatā paritto nakhasikhāyaṃ paṃsu
  āropito”ti. “Evameva kho, bhikkhave, ariyasāvakassa
  diṭṭhi­sam­pannassa puggalassa abhisametāvino etadeva bahutaraṃ
  dukkhaṃ yadidaṃ parikkhīṇaṃ pariyādiṇṇaṃ; appamattakaṃ avasiṭṭhaṃ.
  Neva satimaṃ kalaṃ upeti na sahassimaṃ kalaṃ upeti na satasahassimaṃ
  kalaṃ upeti purimaṃ duk­khak­khan­dhaṃ parikkhīṇaṃ pariyādiṇṇaṃ
  upanidhāya yadidaṃ sattak­khat­tuṃ­paramatā. Evaṃ mahatthiyomaha kho,
  bhikkhave, dhammā­bhisa­mayo; evaṃ mahatthiyo
  dhamma­cak­khu­paṭi­lābho” ti.
sattakkhattuṃ: seven times. + paramatā: (in cpds.) the highest
  quality; at the most.

Where is the word "lives" or "lifetimes" found here? 

user8044: The Nakhasikha Sutta in the Samyutta Nikaya, describes the unimaginably large amount of defilements (and thus future suffering and stress) a Sotapanna has removed compared to a normal human being. The Sutta has this to say:
Once the Buddha picked up a little bit of dust with the tip of his fingernail, and asked the bhikkhus, “What do you think, bhikkhus? Which is greater: the little bit of dust I have picked up with the tip of my fingernail, or the soil in this great Earth?”.
Then the Buddha told the bhikkhus that the amount of defilements that a Sotapanna has removed can be compared to the soil in the whole Earth, while the amount that he/she has left to remove can be compared to the bit of dust on his fingernail.
This tells that the amount of suffering a Sotapanna has left to be endured within seven future bhava is insignificantly small. This is to say that a Sotapanna will never get to the four lowest realms, and s/he will have only seven future bhava left, and those in the human realm or the realms above it. This is so, if we take Sattakkhattuṃparamatā  means - කෙත් වතු 7 ක් පමණයි පසු කර ගත යුත්තේ. This word has a rhetorical, metaphorical and symbolic meaning -  to mean… “One has to cross only seven more fields of (fenced or unfenced) cultivated land. This is what is meant by Sattakkhattuṃparamatā  means - කෙත් වතු 7 ක් පමණයි පසු කර ගත යුත්තේ. This word has a rhetorical, metaphorical and symbolic meaning -  to mean… “One has to cross only seven more fields of (fenced or unfenced) cultivated land. This is what is meant by Sattakkhattuṃparamatā to mean seven bhava the most.
What is the correct meaning of Sattakkhattuṃparamatā. Give me a couple of months the most, and I will give you the deeper meaning of this term. For now let us understand what Sotapanna means, and how could one come to this. 
Out of dasa akusala, only micca ditthi is completely removed at the Sotapanna stage. A Sotapanna would have reduced the “apayagami strength” of most of the other dasa akusala. This is the key to understand this level of attainment. In particular, to get to know that  abhijja or lobha  is reduced to raga level and vyapada or dosa is reduced to patigha level at this stage.
A Sotapanna is said to have achieved “dassanena pahatabba” or removal of defilements via correct vision, with the removal of micca ditthi, or attaining the first stage of Samma Ditthi. Now what is micca ditthi?. My belief as of now is that, if one firmly believes that there is no rebirth or there are no other living beings other than humans and animals, that firm belief is Niyata Micca Ditthi . In finding out the correct meaning of Sattakkhattuṃparamatā, may help in getting to know whether not believing in future bhava is Niyata Micca Ditthi (established wrong views) or not. 
If one has a vessel that is leaking water, there is no point in trying to plug the smaller holes first. One should plug the biggest hole first, which in this case is getting rid of micca ditthi or false views (about this world).
One cannot pretend to believe in things that one truly does not believe in. To just say that “I do believe in rebirth” is one such.  This is a sanna ( “perception”) that one has to be true to. Otherwise it will not work. We “acquire” most sanna through our families first, then through friends, schools, workplaces, etc.
Thus most of our “world views” or ditthis are acquired through our families. Those sannas are hard to change, depending on how forcefully and frequently they have been used. However, human mind is unique. When given enough solid evidence, one’s sanna about something or some concept can permanently change. 
That kind of a “vision change” does not happen quickly, unless one has “sanna” about that from previous lives; that is why it is easier for some people to grasp these concepts.

user10293: Clear and short as needed Buddha said, not to take verbatim. Just to praise the benefit of sotapanna. Sometimes one universe metaphorically said like one palm tree, just like that, suffering in long samsara(long, long duration) just compared to seven lifetime(just  like a wink of eye) at the most of sotapanna. 

user-1: Focusing on pali might not clarify the meaning since it is semantics.  Perhaps other sutta to support and confirm the 7 rebirths (at most) might help you find the answer you are looking for.  From saupadisesa sutta, Buddha classified stream enterer into three categories ,their destinations ,and number of lifetimes;

Ekbiji .  single sprout.  Ekamyeva manusakam bhava nibbattetva (born into human one time and reach nirvana
kolankolo. Two or three times birth.  dve vā tīṇi vā kulāni sandhāvitvā.  Born in clans (human) and/or heaven 2 or 3 times.
sattak­khat­tu­paramo. sattak­khat­tu­paramaṃ deve ca manusse ca sandhāvitvā saṃsaritvā.  Born in heaven or human world 7 times at the most.

Did you notice Buddha used sattakkhattu in both sutta? Tho he didn't explicitly explain 7 lives in Nakhasikhā Sutta, he did so in saupadisesa sutta.  This leaves very little room about the answer to your question for interpretation. 
Buddha had names for three types of stream enterer,  explained their destinations and number of times.
sattakkhattuṃ most likely means seven lifetimes not seven fetters. 
I hope my answer helps.
Saupadisesa sutta

8157: <p>Here is the Pali from the <a href="https://suttacentral.net/pi/sn13.1" rel="nofollow noreferrer">Nakhasikhā Sutta</a>, which is about the "breakthrough" of three fetters, <a href="http://www.accesstoinsight.org/tipitaka/sn/sn13/sn13.001.than.html" rel="nofollow noreferrer">for a disciple of the noble ones who is consummate in view, an individual who has broken through [to stream-entry]</a>:</p>

<blockquote>
  <p>Evaṃ me sutaṃ—​   ekaṃ samayaṃ bhagavā sāvatthiyaṃ viharati jetavane
  anātha­piṇḍi­kassa ārāme. Atha kho bhagavā parittaṃ nakhasikhāyaṃ
  paṃsuṃ āropetvā bhikkhū āmantesi: “taṃ kiṃ maññatha, bhikkhave,
  katamaṃ nu kho bahutaraṃ, yo vāyaṃ mayā paritto nakhasikhāyaṃ paṃsu
  āropito, ayaṃ vā mahāpathavī”ti?</p>
  
  <p>“Etadeva, bhante, bahutaraṃ, yadidaṃ mahāpathavī. Appamattako
  bhagavatā paritto nakhasikhāyaṃ paṃsu āropito. Neva satimaṃ kalaṃ
  upeti na sahassimaṃ kalaṃ upeti na satasahassimaṃ kalaṃ upeti
  mahāpathaviṃ upanidhāya bhagavatā paritto nakhasikhāyaṃ paṃsu
  āropito”ti. “Evameva kho, bhikkhave, ariyasāvakassa
  diṭṭhi­sam­pannassa puggalassa abhisametāvino etadeva bahutaraṃ
  dukkhaṃ yadidaṃ parikkhīṇaṃ pariyādiṇṇaṃ; appamattakaṃ avasiṭṭhaṃ.
  Neva satimaṃ kalaṃ upeti na sahassimaṃ kalaṃ upeti na satasahassimaṃ
  kalaṃ upeti purimaṃ duk­khak­khan­dhaṃ parikkhīṇaṃ pariyādiṇṇaṃ
  upanidhāya yadidaṃ <strong>sattak­khat­tuṃ­paramatā</strong>. Evaṃ mahatthiyomaha kho,
  bhikkhave, dhammā­bhisa­mayo; evaṃ mahatthiyo
  dhamma­cak­khu­paṭi­lābho” ti.</p>
  
  <p>sattakkhattuṃ: seven times. + paramatā: (in cpds.) the highest
  quality; at the most.</p>
</blockquote>

<p>Where is the word "lives" or "lifetimes" found here? </p>
 8044: <p>The Nakhasikha Sutta in the Samyutta Nikaya, describes the unimaginably large amount of defilements (and thus future suffering and stress) a Sotapanna has removed compared to a normal human being. The Sutta has this to say:</p>

<p>Once the Buddha picked up a little bit of dust with the tip of his fingernail, and asked the bhikkhus, “What do you think, bhikkhus? Which is greater: the little bit of dust I have picked up with the tip of my fingernail, or the soil in this great Earth?”.</p>

<p>Then the Buddha told the bhikkhus that the amount of defilements that a Sotapanna has removed can be compared to the soil in the whole Earth, while the amount that he/she has left to remove can be compared to the bit of dust on his fingernail.</p>

<p>This tells that the amount of suffering a Sotapanna has left to be endured within seven future bhava is insignificantly small. This is to say that a Sotapanna will never get to the four lowest realms, and s/he will have only seven future bhava left, and those in the human realm or the realms above it. This is so, if we take <strong>Sattakkhattuṃparamatā</strong>  means - කෙත් වතු 7 ක් පමණයි පසු කර ගත යුත්තේ. This word has a rhetorical, metaphorical and symbolic meaning -  to mean… “One has to cross only seven more fields of (fenced or unfenced) cultivated land. This is what is meant by <strong>Sattakkhattuṃparamatā</strong>  means - කෙත් වතු 7 ක් පමණයි පසු කර ගත යුත්තේ. This word has a rhetorical, metaphorical and symbolic meaning -  to mean… “One has to cross only seven more fields of (fenced or unfenced) cultivated land. This is what is meant by <strong>Sattakkhattuṃparamatā</strong> to mean seven bhava the most.</p>

<p>What is the correct meaning of <strong>Sattakkhattuṃparamatā</strong>. Give me a couple of months the most, and I will give you the deeper meaning of this term. For now let us understand what Sotapanna means, and how could one come to this. </p>

<p>Out of dasa akusala, only micca ditthi is completely removed at the Sotapanna stage. A Sotapanna would have reduced the “apayagami strength” of most of the other dasa akusala. This is the key to understand this level of attainment. In particular, to get to know that  abhijja or lobha  is reduced to raga level and vyapada or dosa is reduced to patigha level at this stage.</p>

<p>A Sotapanna is said to have achieved “dassanena pahatabba” or removal of defilements via correct vision, with the removal of micca ditthi, or attaining the first stage of Samma Ditthi. Now what is <strong>micca ditthi</strong>?. My belief as of now is that, if one firmly believes that there is no rebirth or there are no other living beings other than humans and animals, that firm belief is Niyata Micca Ditthi . In finding out the correct meaning of <strong>Sattakkhattuṃparamatā</strong>, may help in getting to know whether not believing in future bhava is Niyata Micca Ditthi (established wrong views) or not. </p>

<p>If one has a vessel that is leaking water, there is no point in trying to plug the smaller holes first. One should plug the biggest hole first, which in this case is getting rid of micca ditthi or false views (about this world).</p>

<p>One cannot pretend to believe in things that one truly does not believe in. To just say that “I do believe in rebirth” is one such.  This is a sanna ( “perception”) that one has to be true to. Otherwise it will not work. We “acquire” most sanna through our families first, then through friends, schools, workplaces, etc.</p>

<p>Thus most of our “world views” or ditthis are acquired through our families. Those sannas are hard to change, depending on how forcefully and frequently they have been used. However, human mind is unique. When given enough solid evidence, one’s sanna about something or some concept can permanently change. </p>

<p>That kind of a “vision change” does not happen quickly, unless one has “sanna” about that from previous lives; that is why it is easier for some people to grasp these concepts.</p>
 10293: <p>Clear and short as needed Buddha said, not to take verbatim. Just to praise the benefit of sotapanna. Sometimes one universe metaphorically said like one palm tree, just like that, suffering in long samsara(long, long duration) just compared to seven lifetime(just  like a wink of eye) at the most of sotapanna. </p>
 -1: <p>Focusing on pali might not clarify the meaning since it is semantics.  Perhaps other sutta to support and confirm the 7 rebirths (at most) might help you find the answer you are looking for.  From saupadisesa sutta, Buddha classified stream enterer into three categories ,their destinations ,and number of lifetimes;</p>

<ul>
<li>Ekbiji .  single sprout.  Ekamyeva manusakam bhava nibbattetva (born into human one time and reach nirvana</li>
<li>kolankolo. Two or three times birth.  dve vā tīṇi vā kulāni sandhāvitvā.  Born in clans (human) and/or heaven 2 or 3 times.</li>
<li>sattak­khat­tu­paramo. sattak­khat­tu­paramaṃ deve ca manusse ca sandhāvitvā saṃsaritvā.  Born in heaven or human world 7 times at the most.</li>
</ul>

<p>Did you notice Buddha used sattakkhattu in both sutta? Tho he didn't explicitly explain 7 lives in Nakhasikhā Sutta, he did so in saupadisesa sutta.  This leaves very little room about the answer to your question for interpretation. 
Buddha had names for three types of stream enterer,  explained their destinations and number of times.</p>

<p>sattakkhattuṃ most likely means seven lifetimes not seven fetters. </p>

<p>I hope my answer helps.
<a href="https://suttacentral.net/pi/an9.12" rel="noreferrer">Saupadisesa sutta</a></p>
