  We simulate the dynamics of a disordered interacting spin chain subject to a
quasi-periodic time-dependent drive, corresponding to a stroboscopic Fibonacci
sequence of two distinct Hamiltonians. Exploiting the recursive drive
structure, we can efficiently simulate exponentially long times. After an
initial transient, the system exhibits a long-lived glassy regime characterized
by a logarithmically slow growth of entanglement and decay of correlations
analogous to the dynamics at the many-body delocalization transition.
Ultimately, at long time-scales, which diverge exponentially for weak or rapid
drives, the system thermalizes to infinite temperature. The slow relaxation
enables metastable dynamical phases, exemplified by a "time quasi-crystal" in
which spins exhibit persistent oscillations with a distinct quasi-periodic
pattern from that of the drive. We show that in contrast with Floquet systems,
a high-frequency expansion strictly breaks down above fourth order, and fails
to produce an effective static Hamiltonian that would capture the pre-thermal
glassy relaxation.
