We theoretically present the intrinsic limits to electron mobility in the modulation-doped AlGaN/GaN two-dimensional electron gas (2DEG) due to effects including acoustic deformation potential (DP) scattering, piezoelectric scattering (PE), and polar-optic phonon scattering (POP). We find that DE and PE are the more significant limiting factors at intermediate temperatures of 40 K to 250 K, while POP becomes dominant as room temperature is approached. Detailed numerical results are presented for the change of electron mobility with respect to temperature and carrier density. We conclude that these three types of phonon scattering, which are generally determined by the material properties but not the technical processing, are hard limits to the 2DEG mobility.