This is more complicated than I thought. Harry Potter, while 8 films, are all adaptations of specific books so it feels less franchise-y than say James Bond or the MCU. I also agree that a franchise has to be more than a trilogy, so while I'd lean toward including Indiana Jones and Lord of the Rings based strictly on the strength of the first three films, the poor quality of the latter entries makes it hard to view them as successful franchises (while being extremely successful trilogies). Even with Star Wars, I haven't liked anything after the original trilogy.
I, III, & V
None of the sequels come close to the first, but there's a lot of fun to be had. I haven't bothered with the Rob Zombie ones though.
Batman, Batman Returns, Mask of the Phantasm, Begins, Dark Knight
Based on how iconic the first two are and then to revive it 30 years later with one of my favorite films of the last decade.