  Sufficient disorder is believed to localize static and periodically-driven
interacting chains. With quasiperiodic driving by $D$ incommensurate tones, the
fate of this many-body localization (MBL) is unknown. We argue that randomly
disordered MBL exists for $D=2$, but not for $D\geq 3$. Specifically, a
putative two-tone driven MBL chain is neither destabilized by thermal
avalanches seeded by rare thermal regions, nor by the proliferation of
long-range many-body resonances. For $D\geq 3$, however, sufficiently large
thermal regions have continuous local spectra and slowly thermalize the entire
chain. En route, we generalize the eigenstate thermalization hypothesis to the
quasiperiodically-driven setting, and verify its predictions numerically.
Two-tone driving enables new topological orders with edge signatures; our
results suggest that localization protects these orders indefinitely.
