Rheological material functions are used to form our conceptual understanding of a material response. For a nonlinear rheological response, the possible deformation protocols and material measures span a high-dimensional space. Here, we use asymptotic expansions to outline low-dimensional measures for describing leading-order nonlinear responses in large amplitude oscillatory shear (LAOS). This amplitude-intrinsic regime is sometimes called medium amplitude oscillatory shear (MAOS). These intrinsic nonlinear material functions are only a function of oscillatory frequency, and not amplitude. Such measures have been suggested in the past, but here, we clarify what measures exist and give physically meaningful interpretations. Both shear strain control (LAOStrain) and shear stress control (LAOStress) protocols are considered, and nomenclature is introduced to encode the physical interpretations. We report the first experimental measurement of all four intrinsic shear nonlinearities of LAOStrain. For the polymeric hydrogel (polyvinyl alcohol - Borax) we observe typical integer power function asymptotics. The magnitudes and signs of the intrinsic nonlinear fingerprints are used to conceptually model the mechanical response and to infer molecular and microscale features of the material.