user78: Will there ever be a need to create a unit of Bitcoin smaller than a Satoshi?
See also: What is a 'Satoshi'?

user111: 
Will there ever be a need to create a unit of Bitcoin smaller than a Satoshi?

Actually, the correct answer is depended on meaning of "ever" in your question. If you are trying to look far away (e.g. more then one century), the answer can be "Yes, there will."
It can in case Bitcoin would spread as widely as seen by bitcoin evangelists.
Keeping in mind all known mankind history and the idea of bitcoin as "digital gold", we can notice that value of gold is raised constantly. This is a result of simple process: amount of goods and services that people can pay for increased quicker then amount of gold.

user1075: As the Bitcoin Wiki explains on "Coin Destruction":

Bitcoin has 2.1 quadrillion raw units, making up 8 decimals of BTC precision, so the entire network could potentially operate on much less than the full quantity of Bitcoins. If deflation gets to the point where transactions of more than 10 BTC are unheard of, clients can just switch to another unit so that, for example, it shows 10 mBTC rather than 0.01 BTC.
The maximum number of raw units might not be enough if the entire world starts using BTC, but it would not be too difficult to increase precision in that case. The transaction format and version number would be scheduled to change at some particular block number after a year or two, and everyone would have to update by then.

Doing some rough calculations, one can see that it might be the case that complete universal adoption of the Bitcoin could result in equivalent values of 1 (future) satoshi = $1 (current) USD.
  total_bitcoins / population * (satoshis/BTC)
= 21e6           / 7e9        * 1e8
= 300000 satoshis/person


user14: It is very unlikely that anything smaller than a Satoshi will be needed. See the math below.
After a little math provoked by a comment... it's unlikely to ever need a smaller than a satoshi (1/100,000,00 of a BTC).  I calculated about 52500 satoshi's to $1 if all bitcoins have been mined and they are worth about $1900 each.
So even up to $9975000 per bitcoin still puts 1 satoshi at 1 cent.

user5406: TL;DR: If transaction volume explodes, Bitcoin is adopted for micro-payments, and the value of bitcoins increases sufficiently, we will need greater divisibility.
First of all, since there are 100,000,000 satoshi to a bitcoin, obviously one satoshi would be worth $0.01 if one bitcoin were valued at $1 million.
Beside the questionable math in tysat's answer, his answer is based on the assumption that there are no use-cases for payments smaller than a cent.
I don't believe that it is a valid assumption to make:
Next to there being other currencies with subunits of smaller value than $0.01, when one satoshi is worth $0.01, that would also be the smallest possible transaction fee one could attach! This would be prohibitively expensive for any micro-payment based business scheme, such as for example Bitcoin based content paywalls, direct internet traffic payments to ISPs, or other automated micro-payment driven schemes: Paying five cents to read an article, you'd have to add a 20% fee.
Now, such services could possibly move off-chain where further precision could be added, yet that would require users to have accounts for each of those services, and to tie up their resources unnecessarily, while reintroducing counter-party risk; essentially reverting a lot of the key selling points of Bitcoin! 
Assuming that the Bitcoin protocol will develop to scale even beyond the transaction capacity of today's credit card systems as recently suggested by Gavin Andresen, it seems plausible that business ideas will appear to challenge even such transaction capacity. As those would though be prohibited by a minimal transaction fee of $0.01, we could see a push towards even smaller fees. 
Should the above lead to a scenario where somebody finds uses for payments with a value of a tenth of a cent ($0.001), and transaction fees for that should be less than 10% of the payment's value, we'd need smaller subunits if one bitcoin becomes more valuable than $10,000. Since we already have breached $1,000 once, that doesn't sound impossible.

Update 2017-03-23: AFAIU, adding further divisibility to Bitcoin would require a hard fork, and therefore is not something that will simply manifest. If necessary, it would require a massive overhaul of all software currently used in the Bitcoin ecosphere. On the other hand, Lightning actually already implemented support for probabilistic sub-satoshi payments.
Update 2018-05-17: Lightning uses millisatoshi to account for channel balances, adding another three magnitudes of precision. Payouts on the blockchain are rounded as the precision there is only to the satoshi.

78: <p>Will there ever be a need to create a unit of Bitcoin smaller than a Satoshi?</p>

<p>See also: <a href="https://bitcoin.stackexchange.com/questions/114/what-is-a-satoshi">What is a 'Satoshi'?</a></p>
 111: <blockquote>
  <p>Will there <em>ever</em> be a need to create a unit of Bitcoin smaller than a Satoshi?</p>
</blockquote>

<p>Actually, the correct answer is depended on meaning of "<em>ever</em>" in your question. If you are trying to look far away (e.g. more then one century), the answer <em>can</em> be "Yes, there will."</p>

<p>It <em>can</em> in case Bitcoin would spread as widely as seen by bitcoin evangelists.</p>

<p>Keeping in mind all known mankind history and the idea of bitcoin as "digital gold", we can notice that value of gold is raised constantly. This is a result of simple process: amount of goods and services that people can pay for increased quicker then amount of gold.</p>
 1075: <p>As the Bitcoin Wiki <a href="https://en.bitcoin.it/wiki/Weaknesses#Coin_destruction" rel="noreferrer">explains</a> on &quot;Coin Destruction&quot;:</p>
<blockquote>
<p>Bitcoin has 2.1 quadrillion raw units, making up 8 decimals of BTC precision, so the entire network could potentially operate on much less than the full quantity of Bitcoins. If deflation gets to the point where transactions of more than 10 BTC are unheard of, clients can just switch to another unit so that, for example, it shows 10 mBTC rather than 0.01 BTC.</p>
<p><strong>The maximum number of raw units might not be enough if the <em>entire world</em> starts using BTC, but it would not be too difficult to increase precision in that case.</strong> The transaction format and version number would be scheduled to change at some particular block number after a year or two, and everyone would have to update by then.</p>
</blockquote>
<p>Doing some rough calculations, one can see that it might be the case that complete universal adoption of the Bitcoin could result in equivalent values of 1 (future) satoshi = $1 (current) USD.</p>
<pre><code>  total_bitcoins / population * (satoshis/BTC)
= 21e6           / 7e9        * 1e8
= 300000 satoshis/person
</code></pre>
 14: <p>It is very unlikely that anything smaller than a Satoshi will be needed. See the math below.</p>

<p>After a little math provoked by a comment... it's unlikely to ever need a smaller than a satoshi (1/100,000,00 of a BTC).  I calculated about 52500 satoshi's to $1 if all bitcoins have been mined and they are worth about $1900 each.</p>

<p>So even up to $9975000 per bitcoin still puts 1 satoshi at 1 cent.</p>
 5406: <p><em>TL;DR: If transaction volume explodes, Bitcoin is adopted for micro-payments, and the value of bitcoins increases sufficiently, we will need greater divisibility.</em></p>

<p>First of all, since there are 100,000,000 satoshi to a bitcoin, obviously one satoshi would be worth $0.01 if one bitcoin were valued at $1 million.</p>

<p>Beside the questionable math in <a href="https://bitcoin.stackexchange.com/a/213/5406">tysat's answer</a>, his answer is based on the assumption that there are no use-cases for payments smaller than a cent.</p>

<p>I don't believe that it is a valid assumption to make:<br>
Next to there being other currencies with subunits of smaller value than $0.01, when one satoshi is worth $0.01, that would also be the smallest possible transaction fee one could attach! This would be prohibitively expensive for any micro-payment based business scheme, such as for example Bitcoin based content paywalls, <a href="https://bitcoin.stackexchange.com/a/24151/5406">direct internet traffic payments to ISPs</a>, or other automated micro-payment driven schemes: Paying five cents to read an article, you'd have to add a 20% fee.</p>

<p>Now, such services could possibly move off-chain where further precision could be added, yet that would require users to have accounts for each of those services, and to tie up their resources unnecessarily, while reintroducing counter-party risk; essentially reverting a lot of the key selling points of Bitcoin! </p>

<p>Assuming that the Bitcoin protocol will develop to scale even beyond the transaction capacity of today's credit card systems as recently <a href="https://gist.github.com/gavinandresen/e20c3b5a1d4b97f79ac2" rel="noreferrer">suggested by Gavin Andresen</a>, it seems plausible that business ideas will appear to challenge even such transaction capacity. As those would though be prohibited by a minimal transaction fee of $0.01, we could see a push towards even smaller fees. </p>

<p>Should the above lead to a scenario where somebody finds uses for payments with a value of a tenth of a cent ($0.001), and transaction fees for that should be less than 10% of the payment's value, we'd need smaller subunits if one bitcoin becomes more valuable than $10,000. Since we already have breached $1,000 once, that doesn't sound impossible.</p>

<hr>

<p>Update 2017-03-23: AFAIU, adding further divisibility to Bitcoin would require a hard fork, and therefore is not something that will simply manifest. If necessary, it would require a massive overhaul of all software currently used in the Bitcoin ecosphere. On the other hand, Lightning actually already implemented support for probabilistic sub-satoshi payments.</p>

<p>Update 2018-05-17: Lightning uses millisatoshi to account for channel balances, adding another three magnitudes of precision. Payouts on the blockchain are rounded as the precision there is only to the satoshi.</p>
