  Inspired by results of Eskin and Mirzakhani counting closed geodesics of
length $\le L$ in the moduli space of a fixed closed surface, we consider a
similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be
equivalently stated in terms of counting the number of conjugacy classes
(within the mapping class group) of pseudo-Anosovs whose dilitations have
natural logarithm $\le L$. Let $\mathfrak N_r(L)$ denote the number of
$Out(F_r)$-conjugacy classes of fully irreducibles satisfying that the natural
logarithm of their dilatation is $\le L$. We prove for $r\ge 3$ that as
$L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$)
lower and upper bounds. These bounds reveal behavior not present in the surface
setting or in classical hyperbolic dynamical systems.
