Methane, the principal component of natural gas, is an important energy source and raw material for chemical reactions. It also plays a significant role in planetary physics, being one of the major constituents of giant planets. Here, we report measurements of the molecular self-diffusion coefficient of dense supercritical CH4 reaching the freezing pressure. We find that the high-pressure behaviour of the self-diffusion coefficient measured by quasi-elastic neutron scattering at 300 K departs from that expected for a dense fluid of hard spheres and suggests a density-dependent molecular diameter. Breakdown of the Stokes-Einstein-Sutherland relation is observed and the experimental results suggest the existence of another scaling between self-diffusion coefficient D and shear viscosity eta, in such a way that D eta/rho =constant at constant temperature, with rho the density. These findings underpin the lack of a simple model for dense fluids including the pressure dependence of their transport properties. Methane is abundant in the Universe, is an important energy carrier and a model system for fundamental studies. Here, the authors measure the self-diffusion coefficient of supercritical methane at ambient temperature up to the freezing pressure, and find a different behavior than expected based on previous models.