Approaching the theoretically limiting open circuit voltage (V_oc) of solar cells is crucial to optimize their photovoltaic performance. Here, we demonstrate experimentally that nanostructured layers can achieve a fundamentally larger Fermi level splitting, and thus a larger V_oc, than planar layers. By etching tapered nanowires from planar indium phosphide (InP), we directly compare planar and nanophotonic geometries with the exact same material quality. We show that the external radiative efficiency of the nanostructured layer at 1 sun is increased by a factor 14 compared to the planar layer, leading to a 70 mV enhancement in V_oc. The higher voltage arises from both the enhanced outcoupling of photons, which promotes radiative recombination, and the lower active material volume, which reduces bulk recombination. These effects are generic and promise to enhance the efficiency of current record planar solar cells made from other materials as well.