ABSTRACT For a permuton let denote the Shannon entropy of the sampling distribution of on points. We investigate the asymptotic growth of for a wide class of permutons. We prove that if has a nonvanishing absolutely continuous part, then has a growth rate . We show that if is the graph of a piecewise continuously differentiable, measure‐preserving function with bounded derivative, then tends to the Kolmogorov–Sinai entropy of . Using genericity arguments, we also prove the existence of function permutons for which does not converge either after normalizing by or by . We study the sampling entropy of a natural family of random fractal‐like permutons determined by a sequence of i.i.d. choices. It turns out that for every , is heavily concentrated. We prove that the sequence either converges or has deterministic log‐periodic oscillations almost surely, and argue toward the conjecture that in the nondegenerate case, oscillation holds. On the other hand, for a straightforward random perturbation of the model of , we prove the almost sure convergence of .