The aim of this paper is to estimate the irrationality of moduli spaces of
hyperk\"ahler manifolds of types K3[n], Kumn​, OG6, and OG10. We prove
that the degrees of irrationality of these moduli spaces are bounded from above
by a universal polynomial in the dimension and degree of the manifolds they
parametrize. We also give a polynomial bound for the degrees of irrationality
of moduli spaces of (1,d)-polarized abelian surfaces.Comment: 29 pages. Comments are welcome