  We study the emergence and stability of a prethermal phase in an integrable
many-body system subjected to a Fibonacci drive. Despite not being periodic,
Fibonacci drives have been shown to introduce dynamical constraints due to
their self-similar structure, unlike random driving protocols. From
perturbative analysis, this has been argued to result in an exponentially long
prethermal phase in the high frequency limit of driving. Examining higher order
terms in the perturbative expansion, we show that the perturbative description
breaks down eventually in such systems at a finite universal order, which
depends solely on features of the Fibonacci sequence. This leads to an onset of
energy absorption at long time scales for intermediate and low driving
frequencies. Interestingly, in spite of the breakdown of an effective
Hamiltonian in the perturbative analysis, we still observe slow logarithmic
heating time-scales, unlike purely random drives.
