  We replicate a renewal process at random times, which is equivalent to
nesting two renewal processes, or considering a renewal process subject to
stochastic resetting. We investigate the consequences on the statistical
properties of the model of the intricate interplay between the two probability
laws governing the distribution of time intervals between renewals, on the one
hand, and of time intervals between resettings, on the other hand. In
particular, the total number ${\mathcal N}_t$ of renewal events occurring
within a specified observation time exhibits a remarkable range of behaviours,
depending on the exponents characterising the power-law decays of the two
probability distributions. Specifically, ${\mathcal N}_t$ can either grow
linearly in time and have relatively negligible fluctuations, or grow
subextensively over time while continuing to fluctuate. These behaviours
highlight the dominance of the most regular process across all regions of the
phase diagram. In the presence of Poissonian resetting, the statistics of
${\mathcal N}_t$ is described by a unique `dressed' renewal process, which is a
deformation of the renewal process without resetting. We also discuss the
relevance of the present study to first passage under restart and to continuous
time random walks subject to stochastic resetting.
