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 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.Comment: 56 pages. Comments are welcome