There are a lot of longer running series that really start to hit their stride with #3. Friday the 13th Part III is the first one where Jason wears the hockey mask and really establishes the version of Jason that is used in future movies. Likewise, Nightmare on Elm Street 3: Dream Warriors is the one that really codifies Freddy, and sets the tone for the dream sequences and dialogue the series is now known for. And the most interesting one: King Kong vs Godzilla was technically both the third Godzilla and third King Kong movie. Maybe not my favorite, but a solid entry in both series that was instrumental in defining the legacy of both franchises.
Evil Dead 3 is Army of Darkness, which is an amazing movie. Hard to compare it to Evil Dead 1 though, as they're not even in the same genre (ED 2 is kind of a "crossover").
I really liked Die Hard 3 (With a Vengence). Definitely better than 2. Not sure if it's better than 1, though.
If you don't count "Magical Mystery Tour" since it was released for TV, then "Yellow Submarine" was the third Beatles movie. It's my favorite of their films.