Pretty much any film franchise that spawned into a serial television series. That ensures the franchise has a lot of exploration of characters and its concepts. A serial series is so much better than a short feature film that has only time to be a cookie-cutter couple of hours. Which is why when I hate when a television series tries its hand at the big screen; the movie will always be a shallow imitation of something grand and great.
My chosen few:
1. The Terminator
The Terminator, Terminator 2: Judgment Day, Terminator 3: Rise Of The Machines, Terminator: The Sarah Connor Chronicles, Terminator Salvation
2. Hercules And The Amazon Women
(It started out as 5 television movies, then spawned its own series with a 2 spinoff series)
Hercules And The Amazon Women, Hercules And The Lost Kingdom, Hercules And The Circle Of Fire, Hercules In The Underworld, Hercules In The Maze Of The Minotaur, Hercules: The Legendary Journeys, Xena: Warrior Princess, Young Hercules (tv movie), Young Hercules (the series)
3. Buffy: The Vampire Slayer
Buffy: The Vampire Slayer (movie), Buffy: The Vampire Slayer (series), Angel
4. Raiders Of The Lost Ark
Raiders Of The Lost Ark, Indiana Jones And The Temple Of Doom, Indiana Jones And The Last Crusade, The Young Indiana Jones Chronicles, Indiana Jones And The Kingdom Of The Crystal Skull)
5. Back To The Future (Greatest Film Trilogy that stayed as a trilogy and should stay forever as a trilogy)
Back To The Future, Back To The Future Part II, Back To The Future Part III
6. The Evil Dead
The Evil Dead, Evil Dead 2: Dead By Dawn, Army Of Darkness
7. The Matrix (another franchise that should stay as a trilogy)
The Matrix, The Matrix Reloaded, The Matrix Revolutions
8. Night Of The Living Dead
Night Of The Living Dead, Dawn Of The Dead, Day Of The Dead, Land Of The Dead, Diary Of The Dead, Survival Of The Dead
9. Pirates Of The Caribbean
Pirates Of The Caribbean: The Curse Of The Black Peral, Pirates Of The Caribbean: Dead Man's Chest, Pirates Of The Caribbean: At World's End, Pirates Of The Caribbean: On Stranger Tides
10. Spider-Man
Spider-Man, Spider-Man 2, Spider-Man 3
Honorable Mention:
Batman Begins
Batman Begins, The Dark Knight, The Dark Knight Rises
Questionable Entry: (I had this as #5 then realized that technically it was just one movie and a tv series, so I didn't know if that counted. You could look at it as the series being a single sequel to the movie or you could look at the two seasons as two sequels. But I just left out it because I wanted to follow the rules without it being too questionable. I counted Buffy because I would consider Angel at least a second sequel (I do count "spin-offs" as sequels).
Witchblade (live-action)
Witchblade (tv movie), Witchblade (the series)