  We investigate the number entropy $S_N$---which characterizes particle-number
fluctuations between subsystems---following a quench in one-dimensional
interacting many-body systems with potential disorder. We find evidence that in
the regime which is expected to show many-body localization (MBL) and where the
entanglement entropy grows as $S\sim \ln t$ as function of time $t$, the number
entropy grows as $S_N\sim\ln\ln t$, indicating continuing particle transport at
a very slow rate. We demonstrate that this growth is consistent with a relation
between entanglement and number entropy recently established for
non-interacting systems.
