Radiative transfer of energy at the nanometre length scale is of great importance to a variety of technologies including heat-assisted magnetic recording, near-field thermophotovoltaics and lithography. Although experimental advances have enabled elucidation of near-field radiative heat transfer in gaps as small as 20-30 nanometres (refs 4-6), quantitative analysis in the extreme near field (less than 10 nanometres) has been greatly limited by experimental challenges. Moreover, the results of pioneering measurements differed from theoretical predictions by orders of magnitude. Here we use custom-fabricated scanning probes with embedded thermocouples, in conjunction with new microdevices capable of periodic temperature modulation, to measure radiative heat transfer down to gaps as small as two nanometres. For our experiments we deposited suitably chosen metal or dielectric layers on the scanning probes and microdevices, enabling direct study of extreme near-field radiation between silica-silica, silicon nitride-silicon nitride and gold-gold surfaces to reveal marked, gap-size-dependent enhancements of radiative heat transfer. Furthermore, our state-of-the-art calculations of radiative heat transfer, performed within the theoretical framework of fluctuational electrodynamics, are in excellent agreement with our experimental results, providing unambiguous evidence that confirms the validity of this theory for modelling radiative heat transfer in gaps as small as a few nanometres. This work lays the foundations required for the rational design of novel technologies that leverage nanoscale radiative heat transfer.