We study mixing at a double exponential rate on analytic observables. In this regime the spectrum of the associated precomposition operator is trivial, so the question is how much of a map is remembered by the rate itself. For finite Blaschke products of the circle with a fixed point in the unit disk, and for the free semigroup actions they generate, we classify the possible rates: no mixing, exponential, or double exponential with sharp exponent \(\gamma= \log p\), where \(p \ge 2\) is the minimal local degree of the generators at that fixed point. The exponent is not rigid: it is not stable under \(C^1\)-perturbations and does not imply \(C^1\)-conjugacy to affine models. Rigidity holds exactly when the exponent is maximal for the degree: a map of degree \(q\) whose exponent attains \(\log q\) is Möbius conjugate to an affine model. Finally, we discuss the joint spectral radius of the associated precomposition operators and show that in the double exponential regime they generate a quasi-nilpotent non-unital algebra.