Suppose a human being randomly chooses a real number $x$ with $0<x<1$. It seems the probability of choosing $x$ is closely related to the Kolmogorov complexity of $x$. That is, a number like $0.1$ or $0.333...$ which requires little information to specify has a high probability of being selected, while most reals between $0$ and $1$ are so complex they could never be specified by a human being, and thus have zero probability of being selected. Are there any probability distributions that formalize this sort of approach?
There is - universal continuous a priori probability. It's essentially probability measure such that probability of sequence is equal to probability of random Turing machine printing this sequence. Of course probability of number is non-zero only if there is some Turing machine that prints this number, so only countable many numbers has non-zero probability to be selected.