Some irrational numbers are not valid games. For instance, I am sure the expansion of let us say π/4 in binary has 000 as a subsequence somewhere. But that could never happen in a game, because it disallows repetitions of a substring three times in a row.

Ah, you are right! At least wrt the "naive" mapping of irrationals to binary representation. My gut still tells me it's bijective, but the mapping has to be a bit more involved :D