Here is the ignore list I use for managing Altium files in SVN. I'm sure they can be converted to a .gitignore format (if it's even needed) without too much trouble:
Note that I disagree with @KyranF, you do not want to archive the
prjpcbstructure files. They're entirely regenerated every time you compile the project, and there's not really anything in there anyways. They are functionally just build-artifacts, and those should not be committed.
Also, I've had my ass saved a few times by the History stuff, if a few hundred MB of local storage is a problem, you need 1. A bigger hard drive, and 2. To fix your priorities, if a few hundred MB is a serious consideration at all.
I also disagree strongly about committing gerbers. If you're trying to canonically link a actual PCB to a set of files, having the gerbers can be essential, particularly when things go wrong in the gerber export and/or processing stage. Admittedly, you shouldn't be comitting gerber files every day (you shouldn't be bothering to create them daily either), but you should ALWAYS commit (and ideally tag) each set of gerbers you release to manufacturing.
*~$* may be the temporary file indicator for SolidWorks, not Altium.