That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappear in different contexts with high probability.
The algorithmic probability of a sequence x is the sum over the set of all prefix-free programs u that calculate x of 2^-len(u). Think of the shortest description length as a MAP, while the algorithmic probability integrates over the full prior.
A constant which has many short programs can thus have a greater algorithmic probability than another constant with a slightly lower Kolmogorov complexity. To put it back in context, it's possible that a very short program computes some constant, but it's unlikely to be an important mathematical constants. What's particular about mathematical constants is that they keep appearing in many different situations.