We report experimental results on highly supercritical thermal convection in a rapidly rotating hemispherical shell with parabolic gravity. Using silicone oil as the working fluid and an Ekman number $Ek \,{=}\, 4.7 \,{\times}\, 10^{-6}$ we reach Rayleigh numbers up to $1.2 \,{\times}\, 10^{10}$, over 600 times critical. In-situ temperature measurements show that, at these highly supercritical states where convective heat transfer becomes dominant, the time-averaged temperature in the fluid becomes nearly uniform except in a thin thermal boundary layer near the inner spherical boundary. Heat transfer measurements show that Nusselt number $\hbox{\it Nu}$ increases with Rayleigh number $\hbox{\it Ra}$ as $\hbox{\it Nu} \,{\propto}\, Ra^{0.4}$. The measured amplitudes of temperature fluctuations scale well with a model of geostrophic convective turbulence. We also examine convection in a two-layer fluid in the same geometry, using layers of water and silicone oil to produce a stable density stratification. We determine the dependence of heat transfer on the thickness ratio of the layers.