Using aluminum nitride photonic-chip waveguides, we generate optical-frequency-comb supercontinuum spanning from 500 to 4000 nm with a 0.8-nJ seed pulse, and we show that the spectrum can be tailored by changing the waveguide geometry. Since aluminum nitride exhibits both quadratic and cubic nonlinearities, the spectra feature simultaneous contributions from numerous nonlinear mechanisms: supercontinuum generation, difference-frequency generation, second-harmonic generation, and third-harmonic generation. As one application of integrating multiple nonlinear processes, we measure and stabilize the carrier-envelope-offset frequency of a laser comb by direct photodetection of the output light. Additionally, we generate approximately 0.3 mW of broadband light in the 3000- and 4000-nm spectral region, which is potentially useful for molecular spectroscopy. The combination of broadband light generation from the visible through the midinfrared, combined with simplified self-referencing, provides a path towards robust comb systems for spectroscopy and metrology in the field.