Cluster varieties are geometric objects corresponding to cluster algebras; they have many open subsets called cluster tori. These tori cover almost all of the cluster variety, but not quite all of it; the "deep locus" is the part of the cluster variety which is not in any cluster torus. In joint work with Marco Castronovo, Mikhail Gorsky and José Simental Rodríguez, we conjecture a description of the deep locus, and prove it for braid varieties on 2 and 3 strands. In this talk, I will explain our conjecture, and I will make clear what combinatorial problem we'd need to solve in order to prove this result for all braid varieties. I will not assume that the audience has seen cluster varieties or braid varieties before and, indeed, I hope that this talk will serve as a good introduction to those topics.