  We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler
varieties deformation equivalent to the Hilbert scheme of points on a K3
surface or to O'Grady's ten dimensional variety. This question was studied by
Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the
divisibility of the polarization is one. We generalize their results to higher
dimension and divisibility. As a main result, for almost all dimensions $2n$ we
provide a lower bound on the degree such that for all higher degrees, every
component of the moduli space of polarized hyperk\"ahler varieties of
$K3^{[n]}$ type is of general type.
