Thinking about the ever-fascinating Pi on 3/14 ....
If we wait long enough, will any finite subsequence of digits of pi repeat themselves an infinite number of times? Would this be true of any transcendental number, or any irrational number? and what is the expected repeat time? does it depend on the number in any way? My guess is yes, but the details might say something about the distribution of the digits.
A number is said to be normal if every finite sequence of digits of length $n$ occurs in its decimal expansion $\frac{1}{10^n}$ of the time. Borel proved, using the Borel-Cantelli lemma, that almost all (this is a technical term) real numbers are normal. The basic idea is that almost all real numbers behave as if their decimal expansions are chosen randomly. There is no reason not to believe that $\pi$ also behaves more or less like this, but this is a wide open problem.
It's certainly not true that all transcendental numbers behave like this; one can cook up examples like the Liouville constant
$$\sum_{n=1}^{\infty} \frac{1}{10^{n!}}$$
whose only decimal digits are $0$ and $1$, and which was in fact the first number to be proven transcendental.
So the expected repeat time for a finite sequence of $n$ digits is about $10^n$ as it would be for a randomly chosen sequence of digits, not that anyone knows how to prove this.
On the other hand, see this MO question regarding the question of whether this implies that the digits of $\pi$ would make a good pseudorandom number generator.