  In quantum many-body systems, a Hamiltonian is called an ``extensive entropy
generator'' if starting from a random product state the entanglement entropy
obeys a volume law at long times with overwhelming probability. We prove that
(i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive
entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the
non-degenerate gap condition is satisfied almost everywhere. Specializing to
many-body localized systems, these results imply the observation stated in the
title of Bardarson et al. [PRL 109, 017202 (2012)].
