Interesting question. I guess "fun" is pretty subjective!
As both a history and a gun fan, I've enjoyed black powder firearms for many years (I started building "Mississippi" style percussion rifles when I was 13 or 14... I had an uncle who was a muzzleloader and bow hunter, and he introduced me to the craft.)
Obviously, in comparison with modern guns, they're challenging to shoot. One kind of works on the gun between shots, trying to get more accuracy, like removing the barrel and filing the stock to get the barrel to seat better and so forth. If "maintenance" can bond you to your weapon, well... there's plenty of bonding to be had.
Breaking an old rifle down, sanding and reblueing the barrel, polishing the brass furniture & lockplate, staining the stock and giving it a through going over with beeswax until it glows with a rich hue, and then reassembly, all makes for a good afternoon. The barrels *definitely* rust, even stored indoors in a humidity-controlled environment, and a flintlock or musket will have you appreciating oilcloth in a way modern guns do not.
Kind of like old sports cars - yes, they smell like lawnmowers and break down all the time, but if you *really* want to understand a car, you need to experience one of those finicky bastards. Black powder guns kind of play a similar role for guns.
Specific gun in the category? Maybe model 1841 US percussion rifle, 60 grains, .530 patched ball.
DH