  While a clean driven system generically absorbs energy until it reaches
`infinite temperature', it may do so very slowly exhibiting what is known as a
prethermal regime. Here, we show that the emergence of an additional
approximately conserved quantity in a periodically driven (Floquet) system can
give rise to an analogous long-lived regime. This can allow for non-trivial
dynamics, even from initial states that are at a high or infinite temperature
with respect to an effective Hamiltonian governing the prethermal dynamics. We
present concrete settings with such a prethermal regime, one with a
period-doubled (time-crystalline) response. We also present a direct diagnostic
to distinguish this prethermal phenomenon from its infinitely long-lived
many-body localised cousin. We apply these insights to a model of the recent
NMR experiments by Rovny et al., [Phys. Rev. Lett. 120, 180603 (2018)] which,
intriguingly, detected signatures of a Floquet time crystal in a clean
three-dimensional material. We show that a mild but subtle variation of their
driving protocol can increase the lifetime of the time-crystalline signal by
orders of magnitude.
