  Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order
equal to the Moore bound $M(d,k) = 1 + d + d^2 + \dots +d^k$, arise in the
study of optimal network topologies. In an attempt to find digraphs with a
`Moore-like' structure, attention has recently been devoted to the study of
small digraphs with minimum out-degree $d$ such that between any pair of
vertices $u,v$ there is at most one directed path of length $\leq k$ from $u$
to $v$; such a digraph has order $M(d,k)+\epsilon $ for some small excess
$\epsilon $. Sillasen et al. have shown that there are no digraphs with
out-degree two and excess one. The present author has classified all digraphs
with out-degree two and excess two. In this paper it is proven that there are
no diregular digraphs with out-degree two and excess three for $k \geq 3$,
thereby providing the first classification of digraphs with order three away
from the Moore bound for a fixed out-degree.
