  Let $S$ be a closed orientable hyperbolic surface, and let $\mathcal{O}(K,S)$
denote the number of mapping class group orbits of curves on $S$ with at most
$K$ self-intersections. Building on work of Sapir [16], we give upper and lower
bounds for $\mathcal{O}(K,S)$ which are both exponential in $\sqrt{K}$.
