  Rovibrational quantum states in the $X^1\Sigma_g^+$ electronic ground state of H$_2$ are prepared in the $v=13$ vibrational level up to its highest bound rotational level $J=7$, and in the highest bound vibrational level $v=14$ (for $J=1$) by two-photon photolysis of H$_2$S. These states are laser-excited in a subsequent two-photon scheme into $F^1\Sigma_g^+$ outer well states, where the assignment of the highest ($v,J$) states is derived from a comparison of experimentally known levels in \F, combined with \emph{ab initio} calculations of \X\ levels. The assignments are further verified by excitation of $F^1\Sigma_g^+$ population into autoionizing continuum resonances which are compared with multi-channel quantum defect calculations. Precision spectroscopic measurements of the $F-X$ intervals form a test for the \emph{ab initio} calculations of ground state levels at high vibrational quantum numbers and large internuclear separations, for which agreement is found. 
Photolysis production and spectroscopic investigation of the highest   vibrational states in H$_2$ (X$^1\Sigma_g^+$ $v=13,14$)