Possibly, possibly not. You would need to periodically add new files and stage changes (git user talking) and commit. Whether this is what you would want or not is another question.
Perhaps one approach might be to have git add/commit every hour or so if there were changes made. You could write a script/set of scripts which handle the check for changes/add/commit processes and run it with a scheduled task for your user every hour. If there are no changes, your script should be sure not to try and commit.
I'm not sure this is an optimal solution, and you will end up using more and more disk space, particularly if you often store a lot of files (will this contain your 'My Music' folder for example?).