One would expect a repdigit sequence of length k to appear once in a 10^k normal sequence. So in the 10^1000000 expansion of pi, one should expect a million zeroes to appear in a row once. You can check this to a limited extent here: http://www.angio.net/pi/. It only goes up to 200 million (2 x 10^8) digits, but sure enough 00000000 happens to appear twice. (Nonetheless, we don't actually know if pi is normal.)
Thanks for this, very interesting. My earlier intuition is more than confirmed; If you turn every atom in the universe into a supercomputer, get them all working together generating digits of pi, for a million universe life times, you only get 10^80 (atoms in universe approx) * 10^10 (10 billion digits per sec each say) * 10^11 (years in a universe life time roughly) * 10^8 (seconds in a year roughly) * 10^6 (a million universe lifetimes) = 10^115 (ish) digits. In fact that is an incomparably smaller number that 10^1000000 digits, only good for a single run of 115 zeroes in a row.
Basically there's an awful lot of room between zero and mathematical infinity, and normal large numbers that we experience or even try to reason about in everyday life don't and can't begin to stretch even a little way towards the ultimate mathematical limits.
