I remember renting FotR when I was sick and it was relatively new, and since I had nothing to do I watched it over and over and over. It never got boring for me -- I was new to LotR and thought the whole movie was fascinating and exciting. I know I could not do the same thing with The Two Towers, simply because it's that awkward middle stage, as has already been mentioned.
It's when it comes to RotK that I disagree with you. I found this movie to be the most beautiful of the three. It brings tears to my eyes, and rarely does a movie have that effect on me. It still doesn't seem old to me even though I don't watch it very much (simply out of practicality due to its length). It's like reading a familiar fairy tale many times over: the dangerous parts still make you feel anxious and the climax still places something in your heart you can't explain no matter how well you know the story.
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