  Periodically driven, or Floquet, disordered quantum systems have generated
many unexpected discoveries of late, such as the anomalous Floquet Anderson
insulator and the discrete time crystal. Here, we report the emergence of an
entire band of multifractal wavefunctions in a periodically driven chain of
non-interacting particles subject to spatially quasiperiodic disorder.
Remarkably, this multifractality is robust in that it does not require any
fine-tuning of the model parameters, which sets it apart from the known
multifractality of $critical$ wavefunctions. The multifractality arises as the
periodic drive hybridises the localised and delocalised sectors of the undriven
spectrum. We account for this phenomenon in a simple random matrix based
theory. Finally, we discuss dynamical signatures of the multifractal states,
which should betray their presence in cold atom experiments. Such a simple yet
robust realisation of multifractality could advance this so far elusive
phenomenon towards applications, such as the proposed disorder-induced
enhancement of a superfluid transition.
