Batteries work by transferring electrons between two metals via an electrolyte. This is a chemical process that generates heat as the electrons flow between the materials. If the flow happens too quickly, the battery will catch fire.
When you charge a battery, you are pushing the electrons back from where they came, but heat is still generated. So if you charge a battery too quickly, you cause too much heat and the battery catches fire.
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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