A hierarchical family of analytical Born-Oppenheimer potential energy surfaces has been developed for the H+H-2 system. Ab initio calculations of near full configuration interaction (FCI) quality (converged to within approximate to1 muE(h)) were performed for a set of 4067 configurations with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets. The complete basis set (CBS) limit energies were obtained using a highly accurate many-body basis set extrapolation scheme. Surfaces were fitted for the estimated CBS limit, as well as for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets using a robust new functional form. The mean unsigned fitting error for the CBS surface is a mere 0.0023 kcal/mol, and deviations for data not included in the fitting process are of similarly small magnitudes. Highly accurate calculations of the saddle point and van der Waals minimum configurations were performed using basis sets as large as aug-mcc-pV7Z, and these data show excellent agreement with the results of the extrapolated potential surface. The remaining errors from fitting, correlation treatment, and basis set incompleteness for the new CBS-limit surface are lower by over an order of magnitude compared to any prior analytic surface, and are all now significantly smaller than non-Born-Oppenheimer effects. We expect that this new family of potential surfaces will prove useful in studies elucidating the sensitivity of dynamical quantities to the quality of the potential surface.