I can't actually remember, as I have had it for about 25 years. I assume I probably replaced it at some point, but batteries in the first Gen games tend to last longer than the later gens, so it might actually be the original.
I used to back up my save and then start new games to help friends collect the starters. After I traded away the new starters, I'd restore my original save and keep playing, so that save has been backed up and restored a number of times along the way.
I'd like to actually get a dumper at some point so that I can more securely save the file, but for the time being it's saved on the same Mega Memory Card I was using back then.
Seconding this. I used a tool called GBxCart RW and it worked a treat. I'd link to it, but I don't want to appear as if I'm advertising it. There are a couple other tools I think; this is simply the one I used.
I also used it to backup some ancient photos on my Gameboy Camera!