I will discuss some recent forays into some counting problems for free objects. I will focus on free inverse semigroups and free regular $*$-semigroups. I will first discuss recent results joint with M. Kambites, N. Szakács, and R. Webb giving a precise rate of exponential growth of the free inverse monoid of arbitrary (finite) rank, which turns out to be given by a surprisingly complicated but algebraic number. I will then discuss a useful tool for counting algebraic things - rewriting systems - and an elegant bijection which proves a surprising result about the rate of growth of the monogenic free regular $*$-semigroup. Then, and again using the theory of rewriting systems, I will discuss just how non-finitely presented some of these free objects are, and some homological corollaries