The variety of Boolean algebras constitutes a famous example.
The free Boolean algebra on n generators has cardinality 22n, consisting of the n-ary operations 2n→2.
The variety of pointed sets constitutes a trivial example: the free pointed set on n generators has cardinality n+1, consisting of the generators along with the basepoint.
Define a graph G = (E,s,t) to be a set E of edges and unary operations s, t of source and target satisfying s(s(e)) = t(s(e)) and s(t(e)) = t(t(e)).
This is not the case for the variety of Boolean algebras or of pointed sets.