[AaronBrown]

<< given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws? >>

I'd say it differently than pzhon. English is not precise enough to make the answer clearly yes or no.

Pick any number N, no matter how big. If you deal 221^N + N - 1 hands, you will get, on average, one sequence of N 99's in a row. If you deal, say, ten times this many hands, you will be virtually certain of getting at least one sequence of N 99's in a row. So as long as you specify N and p first, I can tell you how many hands you have to deal to have probability p of getting a sequence of N 99's in a row, and that answer is always a finite number.

On the other hand, you can name any number M and any p, and I can give you a number N such that there is less than probability p of getting a sequence of N 99's in a row during M deals (in fact, I don't have to think very hard, I can set N = M + 1 for any p).

So the question comes down to whether you name the number in the sequence first or the number of deals first. There is no limit to the longest sequence of 99's in an infinite number of deals, for any number N you name there will be a longer streak, but whether that means there is "a streak of pocket nines that itself goes on for an infinite number of draws" is not a question with a well-defined answer.