  We derive a topological classification of the steady states of
$d$-dimensional lattice models driven by $D$ incommensurate tones. Mapping to a
unifying $(d+D)$-dimensional localized model in frequency space reveals
anomalous localized topological phases (ALTPs) with no static analog. While the
formal classification is determined by $d+D$, the observable signatures of each
ALTP depend on the spatial dimension $d$. For each $d$, with $d+D=3$, we
identify a quantized circulating current, and corresponding topological edge
states. The edge states for a driven wire ($d=1$) function as a quantized,
nonadiabatic energy pump between the drives. We design concrete models of
quasiperiodically driven qubits and wires that achieve ALTPs of several
topological classes. Our results provide a route to experimentally access
higher dimensional ALTPs in driven low-dimensional systems.
