Hylleraas-Configuration Interaction (Hy-CI) method variational calculations with up to 4648 expansion terms are reported for the ground S state of neutral helium. Convergence arguments are presented to obtain estimates for the exact nonrelativistic energy of this state. The nonrelativistic energy is calculated to be -2.9037 2437 7034 1195 9829 99 a.u. Comparisons with other calculations and an energy extrapolation give an estimated nonrelativistic energy of -2.9037 2437 7034 1195 9830(2) a.u., which agrees well with the best previous variational energy, -2.9037 2437 7034 1195 9829 55 a.u., of Korobov, obtained using the universal (exponential) variational expansion method with complex exponents. In addition to He, results are also included for the ground S states of H−, Li, Be, and B.