I won't prove this here: for any digit, there exists an arbitrarily long sequence of the same digit repeating somewhere within infinite decimal expansion of Pi. Further, this happens infinitely throughout the expansion. Thus, when dealing with Pi exactly, this question does not have an answer.

However, since we're dealing with such a limited representation of only a few trillion digits, this may have an answer...

Edit: as @nightcracker points out, this is conjecture. It is not currently know as to whether or not Pi has that property. (I could have sworn I read a proof years back). Thanks for the correction. With that said, it's not know as to whether or not this question has an answer.