We present a powerful new technique for the extrapolation of ab initio data based on many-body decompositions. Using the new methodology and subtle modifications of the standard correlation consistent basis sets, the H+H-2 barrier height is estimated at 9.603 kcal/mol with a precision of about 0.003 kcal/mol; this extremely accurate result is all the more striking as it can be obtained using basis sets no larger than aug-cc-pVQZ. The method is also used to yield highly accurate energies for the H+H-2 system on a grid of points previously calculated by quantum Monte Carlo. The three-body energy, summed with exact one- and two-body energies, is observed to yield a useful approximate lower bound for the total energy. The highly accurate energies afforded by this method can also be used to assess the accuracy of previously calculated data that has been used to construct potential energy surfaces. As an example, we make a detailed comparison between the new results and the quantum Monte Carlo results for H+H-2.