We advance a Bayesian concept of intrinsic asymptotic universality, taking to its final conclusions previous conceptual and numerical work based upon a concept of a reprogrammability test and an investigation of the complex qualitative behaviour of computer programs. Our method may quantify the trust and confidence of the computing capabilities of natural and classical systems, and quantify computers by their degree of reprogrammability. We test the method to provide evidence in favour of a conjecture concerning the computing capabilities of Busy Beaver Turing machines as candidates for Turing universality. The method has recently been used to quantify the number of intrinsically universal cellular automata, with results that point towards the pervasiveness of universality due to a widespread capacity for emulation. Our method represents an unconventional approach to the classical and seminal concept of Turing universality, and it may be extended and applied in a broader context to natural computation, by (in something like the spirit of the Turing test) observing the behaviour of a system under circumstances where formal proofs of universality are difficult, if not impossible to come by.