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High quality factor optical microcavities have been employed in a variety of material systems to enhance nonlinear optical interactions. While single-crystalline aluminum nitride microresonators have recently emerged as a low loss platform for integrated nonlinear optics such as four wave mixing and Raman lasing, few studies have investigated this material for second-harmonic generation. In this letter, we demonstrate an optimized fabrication of dually resonant phase-matched ring resonators from epitaxial aluminum nitride thin films. An unprecendented second-harmonic generation efficiency of 17 000%/W is obtained in the low power regime, and pump depletion is observed at a relatively low input power of 3.5 mW. This poses epitaxial aluminum nitride as the highest efficiency second-harmonic generator among current integrated platforms.