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≥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≥3
Nearby in the stack
, 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.