I just saw this thread and it brings back memories. Hard to believe these movies are already a decade old. Time flies.
Choosing the best of the three is a fairly simple task for me though. First of all, it's a only a contest between The Fellowship of the Ring and The Return of the King as far as I'm concerned; in The Two Towers, the Frodo and Sam storyline was deliberately held back to save the Shelob sequence for the next movie, giving us a meaningless side trip to Osgiliath. Add that to Aragorn's fake death and the unnecessary Arwen scenes and you've got a lot of time-wasting going on. And that's the last thing you need in movies as long as these ones were.
Also, I still can't understand why the Shelob sequence was held off for the third movie. I know Peter Jackson has said that chronologically, it belonged there but I doubt that was his true motive for holding it back. It seems far more likely that he wanted to pack the final act with everything but the kitchen sink and it was the wrong move. The cliffhanger ending the Shelob sequence provides in the books was so wonderfully cinematic, why would an adaption of the story for the cinema eschew it? It makes no sense.
So that leaves The Fellowship of the Ring and The Return of the King but for my money, I'm going to go for the first of these.
I like the more linear narrative. It's not as jumpy. The smaller scope of the story allows you to spend more time with certain characters and it's the only movie of the three that feels self-contained. No, the ring is not destroyed but it tells the tale of the formation of the fellowship of the ring, the trials and tribulations its members share and ultimately, the breaking of the fellowship.
I also feel it is more atmospheric, especially when it reaches the scenes in the mines and Lothlorien. The only real thing I didn't particular care for was the sickly orange hue to the Rivendell scenes.
An additional point in its favour is that, while all the movies attempted to walk a line somewhere between straightforward adaptations and more cinematic enterprises, I feel that The Fellowship of the Ring is the only movie of the three that did this particularly well.
Finally, of all three, The Fellowship of the Ring is the only movie that I thought had real tension. Although I know Gandalf returns, his fall in Moria is nonetheless wrenching and the impact on the other members of the fellowship feel all too real. And the Urak-hai attack on the surviving members at the end still holds up and is easily the most powerful battle scene of the three movies. It may be the smallest in scale but it's the one the has the most tension because the danger feels real. You can't help thinking that at any moment, Aragorn could get killed while in the other movies, he may as well be Superman. It's also a good example of how the movie balanced fidelity to the source material with the needs of the cinema, giving the story a satisfying ending and depicting a pivotal sequence that was glossed over in the books.
Compare all that to Return of the King, with its ponderous beginning (the ending of The Two Towers transposed), less focused plot lines, almost superhuman protagonists (Legolas, I am looking at you) and bloated self-indulgence, and it's really no contest.