GregoryChaitin, working with AlgorithmicInformationTheory formulated a number called ''Omega'', a number with unsurpassible complexity.   Basically, define the SimplestTuringMachine, iterate through every possible n-bit program as n goes from 1 to inf, output a 1 if it halts, a 0 if it doesn't.  The most difficult serious to compute.
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See also:  GeneralPurposeComputer