  The diameter of a strongly connected $d$-dimensional simplicial complex is
the diameter of its dual graph. We provide a probabilistic proof of the
existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d
\cdot d!} - (\log n)^{-\epsilon}) n^d$. Up to the first order term, this is the
best possible lower bound for the maximum diameter of a $d$-complex on $n$
vertices as a simple volume argument shows that the diameter of a
$d$-dimensional simplicial complex is at most $ \frac{1}{d} \binom{n}{d}$. We
also find the right first-order asymptotics for the maximum diameter of a
$d$-pseudomanifold on $n$ vertices.
