  In contrast to their seemingly simple and shared structure of independence
and stationarity, L\'evy processes exhibit a wide variety of behaviors, from
the self-similar Wiener process to piecewise-constant compound Poisson
processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari
(2018), we characterize their compressibility by studying the entropy of their
double discretization (both in time and amplitude) in the regime of vanishing
discretization steps. For a L\'evy process with absolutely continuous
marginals, this reduces to understanding the asymptotics of the differential
entropy of its marginals at small times, for which we obtain a new local
central limit theorem. We generalize known results for stable processes to the
non-stable case, with a special focus on L\'evy processes that are locally
self-similar, and conceptualize a new compressibility hierarchy of L\'evy
processes, captured by their Blumenthal-Getoor index.
