Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments, and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as 10^{-7}. By adding an analytical parametrization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible with the experimental data. Finally, similarly to nonrotating convection, we find no single scaling behavior, but instead that multiple well-defined dynamical regimes exist in rapidly rotating convection systems.