Is a CMOS battery rechargeable? If so, what does its charger look like?
Most CMOS batteries are CR2032 lithium button cell batteries and are not rechargeable.
There are rechargeable batteries (e.g. ML2032 - rechargeable) that are the same size, but they can't be charged by your computer.
Some motherboards used to have rechargable CMOS batteries on them.
In general now, no CMOS battery (especially the silver button cell type) are rechargable without investing in a specific rechargable version.
Do not - I repeat - DO NOT attempt to recharge a non-rechargable lithium cell (button type) as the resultant explosion and noxious gasses could kill you (or worse).
A standard CMOS battery is not rechargable.
It is a CR20XX type cell, typically a CR2032.
The battery lasts 5-10 years depending on the particular board it is put in. This means it lasts longer than most rechargeable cells would last even being recharged. There is no reason for it to be rechargeable.