  The aim of this paper is to estimate the irrationality of moduli spaces of
hyperk\"ahler manifolds of types K3$^{[n]}$, Kum$_{n}$, 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.
