  The interplay of localizing and delocalizing mechanisms such as disorder,
interactions, and driving is a central topic of modern condensed matter
physics. In this work we experimentally explore the interplay between
quasiperiodic disorder and periodic driving, each of which in isolation is
capable of driving a metal-insulator phase transition. Using a 1D quasiperiodic
cold-atom chain we measure transport across the full phase diagram varying both
drive strength and quasi-disorder strength. We observe lobes of metallic phases
bounded by quantum phase transitions which depend on both drive and disorder.
While these observations are broadly consistent with expectations from a
high-drive-frequency theoretical model, we also observe clear departures from
the predictions of this model, including anomalous changes in localization
behavior at lower drive frequency. We demonstrate experimentally and
theoretically that understanding the full measured phase diagram requires an
extension to commonly-used approximate theories of Floquet matter.
