  Driven quantum systems may realize novel phenomena absent in static systems,
but driving-induced heating can limit the time-scale on which these persist. We
study heating in interacting quantum many-body systems driven by random
sequences with $n-$multipolar correlations, corresponding to a polynomially
suppressed low frequency spectrum. For $n\geq1$, we find a prethermal regime,
the lifetime of which grows algebraically with the driving rate, with exponent
${2n+1}$. A simple theory based on Fermi's golden rule accounts for this
behaviour. The quasiperiodic Thue-Morse sequence corresponds to the $n\to
\infty$ limit, and accordingly exhibits an exponentially long-lived prethermal
regime. Despite the absence of periodicity in the drive, and in spite of its
eventual heat death, the prethermal regime can host versatile non-equilibrium
phases, which we illustrate with a random multipolar discrete time crystal.
