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We simulate the dynamics of a disordered interacting spin chain subject to aquasi-periodic time-dependent drive, corresponding to a stroboscopic Fibonaccisequence of two distinct Hamiltonians. Exploiting the recursive drivestructure, we can efficiently simulate exponentially long times. After aninitial transient, the system exhibits a long-lived glassy regime characterizedby a logarithmically slow growth of entanglement and decay of correlationsanalogous to the dynamics at the many-body delocalization transition.Ultimately, at long time-scales, which diverge exponentially for weak or rapiddrives, the system thermalizes to infinite temperature. The slow relaxationenables metastable dynamical phases, exemplified by a "time quasi-crystal" inwhich spins exhibit persistent oscillations with a distinct quasi-periodicpattern from that of the drive. We show that in contrast with Floquet systems,a high-frequency expansion strictly breaks down above fourth order, and failsto produce an effective static Hamiltonian that would capture the pre-thermalglassy relaxation.

Logarithmically Slow Relaxation in Quasi-Periodically Driven Random Spin Chains