I'm musically knowledgeable, and I love the Amelie soundtrack (composer is Yann Tiersen), Shutter Island soundtrack (various composers), the Pirates of the Caribbean soundtracks (have no idea who composed it), The Piano soundtrack (Michael Nyman), and others that I can't think of.
Those are just movies, though. As far as video games/anime, I absolutely love Yoko Shimomura (Kingdom Hearts and various Final Fantasies), Akira Yamaoka (Silent Hill), Nobuo Uematsu (Final Fantasy VII), Chikayo Fukuda (.hack//G.U.), Yuki Kajiura/See-saw (.hack//SIGN), and Masashi Hamauzu (Final Fantasy XIII and various piano scores).
I agree that what really makes a film, anime, or video game is the music. If the music isn't good, then it makes it very difficult to watch/play. As you can tell, I love a ton of Asian composers, but I think Yann Tiersen is my favorite living composer. He's French, and he composed the soundtrack to Amelie (or "Le Fabuleux Destin D'Amelie Poulain", "The Wonderful Destiny/Life of Amelie Poulain"). It's no coincidence that it's also my favorite movie of all time, despite how odd it is.
EDIT: I forgot to mention Danny Elfman. He composes several of the soundtracks for Tim Burton movies.