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Measurements of the limiting escape rate of atomic hydrogen (H) atoms at Earth, and the relative significance of thermal evaporation and non-thermal escape mechanisms, such as charge exchange and polar wind, have long been lacking. Our recent development of sophisticated radiative transport analysis techniques now enables the reliable interpretation of remotely-sensed measurements of optically-thick H emission, such as those acquired along the Earth\textquoterights limb by the Global Ultraviolet Imager (GUVI) onboard the NASA TIMED spacecraft, in terms of physical parameters such as exobase density and, crucially, vertical diffusive flux. In this work, we present results from a systematic investigation of H Ly_α emission measured by TIMED/GUVI along the Earth\textquoterights dayside limb from 2002-2007, which we use to derive the vertical H flux and associated density distribution from 250 km out to 1 earth radius. Our analysis reveals that the vertical flux of thermospheric H is nearly constant ver a large range of solar activity and typically exceeds the calculated thermal evaporative flux, suggesting that terrestrial H escape is indeed limited by its vertical diffusion. The excess supply of H atoms to the exobase associated with large observed vertical fluxes requires that non-thermal escape mechanisms be operative for steady-state continuity balance. We find that such non-thermal processes are a particularly significant component of total H escape during low solar activity, when thermal evaporation is weakest.