In dynamics, the speed of mixing depends on the dynamical features of the map and the regularity of the observables. Notably, two classical linear models—the Bernoulli doubling map and the CAT map—exhibit double exponential mixing for analytic observables. Are ergodic linear maps the only ones with this property? In dimension one, we provide a full classification for maps from the space of volume-preserving finite Blaschke products acting on the circle (as well as for free semigroup actions generated by a finite collection of such maps). In higher dimensions, we identify a necessary condition for double exponential mixing and present several families of examples and non-examples. Key ideas of the proof involve the Koopman precomposition operator on spaces of hyperfunctions (elements of the dual space of analytic functions), which turns out to be non-self-adjoint, compact, and quasi-nilpotent, with spectrum reduced to zero.