Using atomic force microscopy we have studied the nanomechanical response to nanoindentations of surfaces of highly oriented molecular organic thin films (thickness less than or equal to 1000 nm). The Young's modulus E can be estimated from the elastic deformation using Hertzian mechanics. For the quasi-one-dimensional metal tetrathiafulvalene tetracyanoquinodimethane E similar to 20 GPa and for the a phase of the p-nitrophenyl nitronyl nitroxide radical E similar to2 GPa. Above a few GPa, the surfaces deform plastically as evidenced by discrete discontinuities in the indentation curves associated to molecular layers being expelled by the penetrating tip.