Does pi contain 1000 consecutive zeroes (in base 10)?
Summing up what others have written, it is widely believed (but not proved) that every finite string of digits occurs in the decimal expansion of pi, and furthermore occurs, in the long run, "as often as it should," and furthermore that the analogous statement is true for expansion in base b for b = 2, 3, .... On the other hand, for all we are able to prove, pi in decimal could be all sixes and sevens (say) from some point on.
About the only thing we can prove is that it can't have a huge string of zeros too early. This comes from irrationality measures for pi which are inequalities of the form $|\pi-(p/q)|>q^{-9}$ (see, e.g., Masayoshi Hata, Rational approximations to $\pi$ and some other numbers, Acta Arith. 63 (1993), no. 4, 335-349, MR1218461 (94e:11082)), which tell us that such a string of zeros would result in an impossibly good rational approximation to pi.
A similar question (1 million consecutive 7 in the decimal expansion of pi) has been discussed by Timothy Gowers in a text published in 2006 (see Reuben Hersh: 18 unconventional essays).
His (quite classical) heuristic arguments in favor of yes were even used for a study on the influence of autority on persuasiveness in mathematics (See Matthew Inglis and Juan Pablo Mejia-Ramos, Cognition and Instruction Journal, Routledge, 2009).
Mahler's paper [1] shows that you can't have too many zeros too soon (or any other string of identical digits, or anything else that would give too good of a rational approximation). In this context the result is weak: there's no string of 1000 zeros starting after the 1000/41 st digit of pi... but it's easy enough to calculate 24 digits of pi as it is.
Even assuming that Salikhov's result holds for numbers this small, it won't exclude more than 150 digits (that is, no spans of 1000 zeros in the first 1150 decimal digits of pi).
[1]: K. Mahler, On the approximation of π, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series A, Mathematical Sciences 56 (1953), pp. 30-42.