  Few level quantum systems driven by $n_\mathrm{f}$ incommensurate fundamental
frequencies exhibit temporal analogues of non-interacting phenomena in
$n_\mathrm{f}$ spatial dimensions, a consequence of the generalisation of
Floquet theory in frequency space. We organise the fundamental solutions of the
frequency lattice model for $n_\mathrm{f}=2$ into a quasi-energy band structure
and show that every band is classified by an integer Chern number. In the
trivial class, all bands have zero Chern number and the quasi-periodic dynamics
is qualitatively similar to Floquet dynamics. The topological class with
non-zero Chern bands has dramatic dynamical signatures, including the pumping
of energy from one drive to the other, chaotic sensitivity to initial
conditions, and aperiodic time dynamics of expectation values. The topological
class is however unstable to generic perturbations due to exact level crossings
in the quasi-energy spectrum. Nevertheless, using the case study of a spin in a
quasi-periodically varying magnetic field, we show that topological class can
be realised at low frequencies as a pre-thermal phase, and at finite
frequencies using counter-diabatic tools.
