  In this article, we study the topological Hochschild homology of
$\mathbf{E}_3$-forms of truncated Brown-Peterson spectra, taken relative to
certain Thom spectra $X(p^n)$ (introduced by Ravenel and used by
Devinatz-Hopkins-Smith in the proof of the nilpotence theorem). We prove
analogues of B\"okstedt's calculations $\mathrm{THH}(\mathbf{F}_p) \simeq
\mathbf{F}_p[\Omega S^3]$ and $\mathrm{THH}(\mathbf{Z}_p) \simeq
\mathbf{Z}_p[\Omega S^3\langle{3}\rangle]$. We also construct a topological
analogue of the Sen operator of Bhatt-Lurie-Drinfeld, and study a higher
chromatic extension. The behavior of these "topological Sen operators" is
dictated by differentials in the Serre spectral sequence for
Cohen-Moore-Neisendorfer fibrations.
