  One can define the complexity of a smooth 4-manifold as the minimal sum of
the number of disks, strands and crossings in a Kirby diagram. Martelli proved
that the number of homeomorphism classes of complexity less than n grows as
$n^2$. In this paper we prove that the number of diffeomorphism classes grows
at least as fast as $n^{c\sqrt[3]{n}}$. Along the way we construct complete
kirby diagrams for a large family of knot surgery manifolds.
