  We study the existence of nontrivial and of representable (dual) weak
complementations, along with the lattice congruences that preserve them, in
different constructions of bounded lattices, then use this study to determine
the finite (dual) weakly complemented lattices with the largest numbers of
congruences, along with the structures of their congruence lattices. It turns
out that, if $n\geq 7$ is a natural number, then the four largest numbers of
congruences of the $n$--element (dual) weakly complemented lattices are:
$2^{n-2}+1$, $2^{n-3}+1$, $5\cdot 2^{n-6}+1$ and $2^{n-4}+1$. For smaller
numbers of elements, several intermediate numbers of congruences appear between
the elements of this sequence. After determining these numbers, along with the
structures of the (dual) weakly complemented lattices having these numbers of
congruences, we derive a similar result for weakly dicomplemented lattices.
