  We report heat transfer and temperature profile measurements in laboratory
experiments of rapidly rotating convection in water under intense thermal
forcing (Rayleigh number $Ra$ as high as $\sim 10^{13}$) and unprecedentedly
strong rotational influence (Ekman numbers $E$ as low as $10^{-8}$).
Measurements of the mid-height vertical temperature gradient connect
quantitatively to predictions from numerical models of asymptotically rapidly
rotating convection, separating various flow phenomenologies. Past the limit of
validity of the asymptotically-reduced models, we find novel behaviors in a
regime we refer to as rotationally-influenced turbulence, where rotation is
important but not as dominant as in the known geostrophic turbulence regime.
The temperature gradients collapse to a Rayleigh-number scaling as $Ra^{-0.2}$
in this new regime. It is bounded from above by a critical convective Rossby
number $Ro^*=0.06$ independent of domain aspect ratio $\Gamma$, clearly
distinguishing it from well-studied rotation-affected convection.
