  We consider cosmology with an inflaton scalar field with an additional
quartic kinetic term. Such a theory can be motivated by Palatini $R+R^2$
modified gravity. Assuming a runaway inflaton potential, we take the Universe
to become dominated by the kinetic energy density of the scalar field after
inflation. Initially, the leading kinetic term is quartic and we call the
corresponding period hyperkination. Subsequently, the usual quadratic kinetic
term takes over and we have regular kination, until reheating. We study, both
analytically and numerically, the spectrum of primordial gravitational waves
generated during inflation and re-entering the horizon during the subsequent
eras. We demonstrate that the spectrum is flat for modes re-entering during
radiation domination and hyperkination and linear in frequency for modes
re-entering during kination: kinetic domination boosts the spectrum, but
hyperkination truncates its peak. As a result, the effects of the kinetic
period can be extended to observable frequencies without generating excessive
gravitational waves, which could otherwise destabilise the process of Big Bang
Nucleosynthesis. We show that there is ample parameter space for the primordial
gravitational waves to be observable in the near future. If observed, the
amplitude and `knee' of the spectrum will provide valuable insights into the
background theory.
