Super Omega
As mentioned above, the first n bits of Gregory Chaitin's constant Omega are random or incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O(1) bits. However, consider the short but never halting algorithm which systematically lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first n bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) onto the first n bits of Omega. In other words, the enumerable first n bits of Omega are highly compressible in the sense that they are limit-computable by a very short algorithm; they are not random with respect to the set of enumerating algorithms. Jürgen Schmidhuber (2000) constructed a limit-computable "Super Omega" which in a sense is much more random than the original limit-computable Omega, as one cannot significantly compress the Super Omega by any enumerating non-halting algorithm.
Read more about this topic: Chaitin's Constant