© The Authors 2015. We present laboratory and numerical models investigating the behavioural regimes of rapidly rotating convection in high-latitude planetary core-style settings. Our combined laboratorynumerical approach, utilizing simplified geometries, can access more extreme parameters (e.g. Rayleigh numbers Ra ≲ 1013; Nusselt numbers Nu ≲ 103; Ekman numbers E ≳ 3 × 10-8) than current global-scale dynamo simulations. Using flow visualizations and heat transfer measurements, we study the axialized flows that exist near the onset of rotating convection, as well as the 3-D flows that develop with stronger forcing. With water as the working fluid (Prandtl number Pr ≲ 7), we find a steep scaling trend for rapidly rotating convective heat transfer, Nu~(Ra/RaC)3.6, that is associated with the existence of coherent, axialized columns. This rapidly rotating trend is steeper than the trends found at moderate values of the Ekman number, and continues a trend of ever-steepening scalings as the rotation rate of the system is increased. In contrast, in more strongly forced or lower rotation rate cases, the heat transfer scaling consistently follows a shallower slope equivalent to that of non-rotating convection systems. The steep heat transfer scaling in the columnar convection regime, corroborated by our laboratory flow visualizations, imply that coherent, axial columns have a relatively narrow range of stability. Thus, we hypothesize that coherent convection columns are not stable in planetary core settings,where the Ekman number is estimated to be~10-15. As a consequence, convective motions in the core may not be related to the columnar motions found in presentday global-scale models. Instead, we hypothesize that turbulent rotating convection cascades energy upwards from 3-D motions to large-scale quasi-2-D flow structures that are capable of efficiently generating planetary-scale magnetic fields. We argue that the turbulent regimes of rapidly rotating convection are essential aspects of core dynamics and will be necessary components of robust, next-generation and multiscale dynamo models.