This article, which is substantially motivated by the previous joint work with J. McKay [8], establishes the analytic analogues of the relations we found free probability has with Witt vectors. Therefore, we first present a novel analytic derivation of an exponential map which relates the free additive convolution on $\mathbb{R}$ with the free multiplicative convolution on either the unit circle or $\mathbb{R}_+^*$, for compactly supported, freely infinitely divisible probability measures. We then introduce several new operations on these measures, which gives rise to more extended classes of operations. Then we consider the relation with classical infinite divisibility and the geometry of the spaces involved. Finally, we discuss the general structure, using the language of operads and algebraic theories, of the various operations we defined give rise to.