Driven quantum systems may realize novel phenomena absent in static systems, but driving-induced heating can limit the timescale 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≥1, 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 behavior. The quasiperiodic Thue-Morse sequence corresponds to the n→∞ 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 nonequilibrium phases, which we illustrate with a random multipolar discrete time crystal.