  Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of
vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$,
there exists $\eta_r>0$ such that
$\text{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. Our techniques also allow us to
bound some extremal quantities related to the stationary distribution of a
simple random walk on $D(n,r)$. In particular, we determine the asymptotic
behaviour of $\pi_{\max}$ and $\pi_{\min}$, the maximum and the minimum values
of the stationary distribution. We show that with high probability $\pi_{\max}
= n^{-1+o(1)}$ and $\pi_{\min}=n^{-(1+\eta_r)+o(1)}$. Our proof shows that the
vertices with $\pi(v)$ near to $\pi_{\min}$ lie at the top of "narrow, slippery
towers", such vertices are also responsible for increasing the diameter from
$(1+o(1))\log_r n$ to $(1+\eta_r+o(1))\log_r{n}$.
