For an invariant probability measure for the Gauss map, almost all numbers are Diophantine 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. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''.