Young's moduli of selected amino acid molecular crystals were studied both experimentally and computationally using nanoindentation and dispersion-corrected density functional theory. The Young modulus is found to be strongly facet-dependent, with some facets exhibiting exceptionally high values (as large as 44 GPa). The magnitude of Young's modulus is strongly correlated with the relative orientation between the underlying hydrogen-bonding network and the measured facet. Furthermore, we show computationally that the Young modulus can be as large as 70-90 GPa if facets perpendicular to the primary direction of the hydrogen-bonding network can be stabilized. This value is remarkably high for a molecular solid and suggests the design of hydrogen-bond networks as a route for rational design of ultra-stiff molecular solids.