  We compute the rate of exponential growth of the free inverse monoid of rank
$r$ (and hence an upper bound on the corresponding rate for all $r$-generated
inverse monoids and semigroups). This turns out to be an algebraic number
strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the
rank tends to infinity. We also find an explicit expression for the exponential
growth rate of the number of idempotents, and prove that this tends to
$\sqrt{e(2k-1)}$ as $k \to \infty$.
