unfortunately I don't know a trilogy that I feel is deserving of this honor..

LotR - (I'm a big fan of the books) doesn't do it for me in movie 2+3, in the big battles there is just sth feeling off imo
Back to the Future - consistently good-very good, but individually the movies aren't up there
Toy Story - same as above, I was really surprised how genuinely entertaining the third one was considering how stupid Shrek and Ice Age became the more movies they put out for those, yet not enough overall
original Star Wars trilogy - looking back they've lost too much of their fascination to me
new Star Wars trilogy - lol, no
Batman - I've yet to watch TDKR, but TDK already was a step down from Batman Begins imo, so I don't expect it to turn that around tbh
Matrix - no
Godfather - 1 and 2 are excellent, but I've yet to watch the 3rd one as most people rate that one a few steps below the first 2

I think TV shows just do a better job at maintaining a consistenly superb quality while telling a longer story.

edit:

I guess I misread the question a bit, so just as conclusion movies probably Toy Story 3 and Back to the Future 3 tied for the top

Toy Story 3 easily imo

Indiana Jones if you count only the original 3... but it's not a trilogy anymore.

Didn't like the last 2nd lord of the rings so the third one left me with a meh feeling.
Return of the Jedi, worst of the original trilogy by far, and imo the 2nd worst star wars movie.
Dark Knight Rises, worst of the trilogy.
Matrix, I liked it better then reloaded, but still a step down from 1.

batman the best film of all time? i dont know whether to laugh at you, pity you or curse at you

Return of the King Extended Edition! 4 hours 10 minutes of godly film-making, story telling and epicness. You can't top that!

Toy Story 3 pretty much fulfilled the story perfectly. Toy Story was a series about growing up and both the beauty and pain of moving on, and the ending perfectly encapsulated that.

Out of the movies I've seen on that list, my top five would go like this...

Return of the King
Toy Story 3
Return of the Jedi
Revenge of the Sith
Nothing

HP doesn't count as it's not even close to a trilogy, and the movie was awful anyway. Transformers: Dark of the Moon (which I saw mentioned in the thread) isn't on the list, because it was so terrible that I couldn't get past the first half hour of it.

Toy Story 3 and Return of the King are essentially tied. They're both fantastic movies, and among my all time favourites.

I'll go with:
Return of the King
The Dark Knight Rises
Toy Story 3
Red Dragon
Matrix Revolution
Revenge of the Sith

I haven't seen either of the Star Wars movies but it doesn't matter. Toy Story 3. I'm absolute on this!

WHY??? It's fucking STAR WARS!

Just saying, you would not regret it!

OT:

1. Star Wars: Revenge of the Sith

2. Lord of the Ring: Return of the King

3. Star Wars: Return of the Jedi (stupid ewoks...)

4. The Dark Knight Rises

5. Hmm... Uh... Well... The Naked Gun 3, I guess..?

I've always actually wanted to watch the series, I just never get around to it.