An electronic device may have a power system with a battery. The power system receives power such as wireless power or wired power and uses a portion of the received power to charge the battery as needed. Control circuitry in the portable electronic device is used to run background processes such as image processing tasks, data synchronization tasks, indexing, and other background processes. In some circumstances, such as when the battery is below a certain state of charge threshold, background processes may be stopped so that the battery is charged as fast as possible. Once above this initial state of charge threshold, background processes may be performed during charging as long as the temperature and state of charge of the battery do not exceed safety temperature and safety state of charge values. Performing background processes in these conditions ensures requisite background processing tasks are completed while preserving battery health.