A 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface $S$ of genus $g\ge 2$, the number $N(L)$ of closed Teichmüller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi\in MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)$ is the "dilatation" or "stretch factor" of $\phi$. We consider an analogous problem in the $Out(F_r)$ setting for the number $N_r(L)$ of $Out(F_r)$ conjugacy classes of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that $N_r(L)$ grows doubly exponentially in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.