You can! In a capacitor. They store energy just in the form of electrons on a surface. They literally charge as fast as you can pump on electrons. The problem is, that severly limits capacity compared to batteries and their voltage drops as they discharge, which is usually undesirable (for power storage, that is, it’s very much desired in other uses)
Batteries store energy in chemical bonds. You can increase charging speed by increasing charging voltage, but that creates too much excess heat from the chemical reactions and also risks shorting the battery, which permanently destroys it.
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