Any modern rechargable battery has a charge controller either built into the battery or into the battery charger that detects when the voltage of the battery reaches optimal level (and therefore is full) and stops the flow of power to prevent overcharging.
In a system without a charge controller (like really old battery powered electronics) overcharging results in the battery being damaged in one way or another, usually wearing them out dramatically faster as the chemistry in the battery is irreversibly changed by the excess energy.
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