We present GaN-based high electron mobility transistors (HEMTs) with a 2-nm-thin InAlN/AlN barrier capped with highly doped n(++) GaN. Selective etching of the cap layer results in a well-controllable ultrathin barrier enhancement-mode device with a threshold voltage of +0.7 V. The n(++) GaN layer provides a 290-Omega/square sheet resistance in the HEMT access region and eliminates current dispersion measured by pulsed IV without requiring additional surface passivation. Devices with a gate length of 0.5-mu m exhibit maximum drain current of 800 mA/mm, maximum transconductance of 400 mS/mm, and current cutoff frequency f(T) of 33.7 GHz. In addition, we demonstrate depletion-mode devices on the same wafer, opening up perspectives for reproducible high-performance InAlN-based digital integrated circuits.