  We study two-terminal transport through two-dimensional periodically driven
systems in which all bulk Floquet eigenstates are localized by disorder. We
focus on the Anomalous Floquet-Anderson Insulator (AFAI) phase, a
topologically-nontrivial phase within this class, which hosts topologically
protected chiral edge modes coexisting with its fully localized bulk. We show
that the unique properties of the AFAI yield remarkable far-from-equilibrium
transport signatures: for a large bias between leads, a quantized amount of
charge is transported through the system each driving period. Upon increasing
the bias, the chiral Floquet edge mode connecting source to drain becomes fully
occupied and the current rapidly approaches its quantized value.
