Chaitin's constant (which is actually a family of constants, one that exists for each programming language) is not computable. It's the probability of a random

Chaitin's constant (which is actually a family of constants, one that exists for each programming language) is not computable. It's the probability of a randomly generated program halting. One can imagine it being of interest, at least among theorists. That said, I struggle to think of other useful non-computable numbers, save contrived examples.