A Jonsson-Tarski algebra is a set X endowed with an isomorphism X~X*X. As observed by Freyd, the category of Jonsson-Tarski algebras is a Grothendieck topos. In particular, one can do algebra, topology and functional analysis inside it, and on doing so, the following objects simply pop out: Cantor space; Thompson's group V; the Leavitt algebra L2; the Cuntz etale groupoid O2; and the Cuntz C^*-algebra O2. In this talk, we explain how this happens, and describe other "self-similar toposes" which capture other kinds of structure of interest to algebraists and analysts.