A power system is provided for a mobile machine. The power system may have a traction motor configured to propel the mobile machine during a driving operation and to generate electricity during a braking operation. The power system may also have an aluminum-air battery connected to the traction motor. The power system may further have a circuit fluidly connected to the aluminum-air battery and configured to circulate electrolyte and aluminum hydroxide particles produced by the aluminum-air battery. The power system may also have a crystallizer connected to the circuit and configured to crystallize and separate the aluminum hydroxide particles from the electrolyte. The power system may also have a heating chamber configured to heat the aluminum hydroxide particles and produce aluminum oxide separated from the electrolyte, wherein the heating chamber is powered by electricity generated by the traction motor during the braking operation.