I've personally chronicled 11 endings for CC, although there may be more. If you have SEEN other endings, please E-mail me. I plan on cross-examining you, so make sure to include a reply address. To the best of my knowledge, there ONLY ending with a variation is the Comic Relief one, and it never makes a difference who's in your active party when you finish the game (besides for two lines of text in the "Real" ending).
Most of you will have gotten this the first time around; beat the final boss after finishing Terra Tower. Do this by beating him like you've beaten any other enemy. The game switches to the credits and the background song "Radical Dreamers", and that's it.
To get this, you have to use the Chrono Cross in the final battle, after using Elements in this order: Yellow, Red, Green, Blue, Black, White. See the last part of the walkthrough for hints on how to pull this off. When you do this, you'll be treated to another sequence, a conversation, and two extra FMVs than the Fake Ending. Note that you can get this ending in New Game+ as soon as you get the Chrono Cross, even before finishing Terra Tower!
New Game+ Endings
As in Chrono Trigger, finishing the game as soon as you can using only Serge (this is almost impossible without the Green Plate, and still tough if you have under 60 or so Stars) sends you to meet all the programmers. They live in a pseudo-Viper Manor; there are two floors and four rooms on each floor. You can check out all the music, or look at how different characters would say the same text. Talk to Solt or Peppor to finish the ending. There's a locked door at the top center of this place which I have no idea how to open; let me know if you find out how. (I haven't seen Hironobu Sakaguchi, so maybe he's behind there...)
Beat the game with at least one party member (other than just Serge) but before the Viper Manor sequence. It involves Serge and Leena in Arni, and Kid in Viper Manor. This is a great "alternate reality" ending.
Beat the game after Kid falls ill after the first Viper Manor sequence. (You can't get it if you already got both Nikki and Razzly.) This involves Serge, Lisa, and Leena in the Termina Element shop, and Solt, Peppor, and Pierre in Fort Dragonia. If you've cured Kid in Korcha's path, you'll also see her in the Element shop scene; if you haven't (or if you're in Macha's path), you'll see her in Viper Manor with Norris and Orcha.
Beat the game as Serge having gotten Nikki and Razzly and cured Kid. It involves a concert aboard the Magical Dreamers ship in Termina. The best New Game+ ending visually.
Beat the game with Lynx, before rescuing Riddel. It involves Lynx and Harle in Marbule, and a scene with Zappa, Fargo, and Radius in the Sea of Eden.
Finish the game immediately after rescuing Riddel but BEFORE going to the Hermit's Hideaway. It involves Viper and friends going after Dark Serge. Scenes take place aboard Viper Manor and the S.S. Invincible.
Not so much an ending as showing what happens behind the scenes in the game (this actually does take place, but you don't see the scene yourself in the game). Beat the game after Harle leaves your party, but before getting the Mastermune, to get this ending. (Note that it makes no difference if you have Lynx or Serge; you'll still get this ending.)
Beat the game after getting the Mastermune, but BEFORE finishing Chronopolis. (Again, you get this whether you have Lynx or Serge.) This is probably the funniest ending there is. It involves Dario and Viper building an orphanage in Viper Manor, and the rather creative way the rest of the gang has to come up with the required funds...
Beat the game after you finish Chronopolis, but before finishing Terra Tower (whether you have the Mastermune or not). It involves the Dragon God, the Dwarf Chieftan, the Sage of Marbule, and Harle in Arni Village.