Alright. Now before we get started, let's take a moment to honor some of the fallen trilogies. The ones that came so close to becoming a consistently great trilogy, but just couldn't get past the third chapter...
Now, some of those are still great just from the power of the first two. But when the third movie in a trilogy is not even close to the level as the others, i can't honestly say it is a great "trilogy". Those are some great dynamic duos, but that's as far as i'll go. So remember, i'm not saying that The Godfather Part I and II aren't great (in fact, that's probably the greatest dynamic duo ever), i'm saying that Part III keeps me from classifying it as a great "trilogy". Just want to be clear on that.
Ok, now let's go through some great trilogies.
There are two main types of trilogies. The first one follows one main hero on a journey over the course of three movies. The actions in the previous movie have consequences in the next. This is the most difficult type of trilogy to keep consistant throughout all three movies, because each movie is bound by the actions of it's predecessors. It's quite a challenge to make a great ending to a story with those restrictions. But it's also the most rewarding type of trilogy when it does end great, because you get to see the three-movie story arc come to a satisfying end.
Examples: LOTR, TDK, Star Wars.
The second type of trilogy is a set of three movies that may or may not have the same characters, and the story in one movie has no effect on the story in the next movie. A good example of this would be Daniel Craig's James Bond movies. Yes, many of the same characters are present in all three films, but every movie is completely self-contained, and there are no consequences of any events of the preceeding film. This type of trilogy is less common because there is no need for an ending, so they usually just keep making them until they stop making money.
Examples: The Dollars Trilogy, Three Colors Trilogy, Indiana Jones.
Now, here is how I rank the top five trilogies of all time from best to worst...
Star Wars Trilogy (Original)
This trilogy comes dangerously close to landing in my "fallen trilogies" list from earlier, because 'Return of the Jedi' is not up to the standard of 'A New Hope' or 'The Empire Strikes Back'. But that being said, even though some of the plot was recycled from the first movie, and Ewoks were somehow very formidable against the Empire, it sticks the landing. Our hero's journey comes to a very satisfying end. The Luke Skywalker/Darth Vader storyline, and Luke's relationships with all of the other characters conclude in the best way possible. So yes, this trilogy does have an obvious weak link, but it's a weak link with a satisfying ending. In terms of quality, Star Wars is pretty much overall tied with TDK Trilogy for me. But I give Star Wars the #1 spot simply because of it's historical importance.
Critical/audience ratings:
A New Hope:
94%, 8.2/10 average rating Rottentomatoes
8.8 IMDb
The Empire Strikes Back:
97%, 8.6/10 average rating Rottentomatoes
8.8 IMDb
Return of the Jedi:
79%, 6.8/10 average rating Rottentomatoes
8.4 IMDb
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The Dark Knight Trilogy
Finally, a CBM trilogy can stand among the all-time greats. Spider-Man came close, X-Men came close, and Iron Man already blew it with IM2. People were wondering if they would ever get a great CBM trilogy. Well here it is. The Dark Knight Trilogy. And not only is it a great comic book trilogy, it's one of the greatest of all time by any standard. As it's been noted time and time again, it transcends "comic book" and becomes truly great cinema. Bruce Wayne's journey throughout these three movies is one of the most complex, compelling, and satisfying journeys of any hero ever put on film. And unlike Star Wars, it does not have a weak spot. All three movies were met with great critical and audience reception. Now, if we were forced to pick a "weak link" from the consensus, it would have to be Batman Begins, based off the critic and audience scores. But there is no shame in being seen as the "worst" of TDK Trilogy.
Critical/audience ratings:
Batman Begins:
85%, 7.7/10 average rating Rottentomatoes
70/100 Metacritic
8.3 IMDb
The Dark Knight:
94%, 8.4/10 average rating Rottentomatoes
82/100 Metacritic
9.0 IMDb
The Dark Knight Rises:
87%, 8.0/10 average rating Rottentomatoes
78/100 Metacritic
8.8 IMDb
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The Lord of the Rings Trilogy
The critical acclaim and audience reception of this trilogy is probably the most impressive of all. Each movie was nominated for Best Picture at the Oscars (with The Return of the King winning), and they are all loved by audiences worldwide. I've never read the books, and I was never really a fan of the LOTR culture, so I probably have less appreciation for it than most people do. But I still admire the films themselves, and recognize the trilogy as one of the most epic achievements in film history.
Critical/audience ratings:
The Fellowship of the Ring:
92%, 8.1/10 average rating Rottentomatoes
92/100 Metacritic
8.8 IMDb
The Two Towers:
96%, 8.4/10 average rating Rottentomatoes
88/100 Metacritic
8.7 IMDb
The Return of the King:
94%, 8.6/10 average rating Rottentomatoes
94/100 Metacritic
8.9 IMDb
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Indiana Jones Trilogy
This is one of those trilogies where all three movies are self-contained, and there is no definite ending. There is no "epic journey", no "realizations", and nothing comes "full circle". And yet, it's one of the greatest movie series of all time, and it created one of the most iconic characters ever, Indiana Jones. Some people try to slag off Temple of Doom as if it keeps this from becoming a perfect trilogy (Side note: "Perfect trilogy" does not mean that every movie in the trilogy is perfect). Yes, i agree that Temple of Doom is the weakest movie in the trilogy, but so what? Just because it's the weakest of the three Indiana Jones movies doesn't mean it's not great in it's own right.
Critical/audience ratings:
Raiders of the Lost Ark:
95%, 9.0/10 average rating Rottentomatoes
8.7 IMDb
Temple of Doom:
85%, 7.2/10 average rating Rottentomatoes
7.6 IMDb
The Last Crusade:
89%, 7.9/10 average rating Rottentomatoes
8.3 IMDb
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The Dollars Trilogy
Another trilogy with three self-contained movies. No epic three-movie story ark, just three great movies. Clint Eastwood is the Man With No Name, with his iconic poncho, hat, and cigar. Blazing a trail of badass everywhere he goes.
Critical/audience scores:
A Fistful of Dollars:
98%, 8.1/10 average rating Rottentomatoes
8.0 IMDb
For a Few Dollars More:
94%, 7.8/10 average rating Rottentomatoes
8.3 IMDb
The Good, the Bad, and the Ugly:
97%, 8.8/10 average rating Rottentomatoes
9.0 IMDb
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Honerable mention: Toy Story, Back to the Future, The Matrix, and Evil Dead.