  Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by
the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of
Mathew, Noel, and Naumann, we investigate the extent to which this criterion
holds in the homotopy of $\mathbb{E}_n$-ring spectra. For all odd primes $p$
and all chromatic heights $h$, we use the Cohen-Moore-Neisendorfer theorem to
construct examples of $K(h)$-local, $\mathbb{E}_{2n-1}$-algebras with
non-nilpotent $p^n$-torsion. We exploit the interaction of the Bousfield-Kuhn
functor on odd spheres and Rezk's logarithm to show that our bound is sharp at
height $1$, and remark on the situation at height $2$.
