Diophantine approximation of irrationals is connected with the asymptotics  of the Gauss transformation. For an invariant probability measure for the Gauss map, almost all numbers are Diophantine (i.e., irrationality exponent=2) if the log of the partial quotient function is integrable. We show that with respect to a continued fraction mixing measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville (i.e., irrationality exponent infinite). Time permitting, we'll also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent and prove a Khinchin-type theorem for weak Renyi measures.

The proofs are via  the "extravagance'' of non-negative stationary processes.

Work in progress with Hitoshi Nakada. See arXiv:2409.19393 for preliminary version.