Its funny, I was actually just thinking about this last weekend while watching some of the ROTK: EE (hope to watch the whole thing tomorrow). Its a tough question, and I do agree with those of you who say that it probably isn't really fair to compaire the two trilogies (LOTR and the Star Wars OT, that is). It is a common comparison, and I can see why it happens, but they really are totally different movies. Lord of the Rings comes from what is regarded as one of the greatest literary works, and an unbelievably expanded backstory on the characters/areas, etc. Star Wars came from the mind of Lucas, borrowing from a number of different areas, and came out great. Although the prequels have seemed to dim the light on Star Wars a little bit for some people, the OT is still an amazing work overall, and I hope people don't forget that. I was kind of young when they were out, but remember watching the first two on tape, and seeing ROTJ in the theatre when it was released. Those movies were just amazing at the time, and have held up really well (even without the SE enhancements).
I like the example used above, the Star Wars OT is just "fun". I really never tire of watching them, and could pop them in any week even though I could probably play the movie through in my mind at this point anyways. These movies will always be tops with me. That said, I didn't think anything would ever come close to the Star Wars experience with me...until I saw the LOTR series in the theatres over the past few years. I don't know that I had ever been so "in" to a movie as when I saw these for the first time. I really just felt immersed in the story and had those experiences where you totally lose the concept of time of day, where you are, just being into the movie. Part of this might have stemmed from the fact that I hadn't read the books beforehand, so it was all new to me. I really respect the LOTR work overall, the outstanding job that Peter Jackson and his crew did with it, the initial story from Tolkien, and as Scott mentioned, the under-rated acting. I can't picture anyone else playing any one of those characters, the actors just all seem "perfect" for their respective roles.
As far as the prequels go, well LOTR blows those out of the water if you are comparing them. But then again, it still isn't a real fair comparison. I did enjoy both of the prequels when I first saw them (hey, its new Star Wars), but will admit that the luster has worn off over time somewhat. I am really looking forward to the next installment, ROTS, and I think a lot of that is because there are even more connections to the OT. The more OT-ish these movies get, it seems the more I enjoy them. I also thought that overall the effects were pretty good in the prequels. Some parts look kind of hokey (especially when Anakin "surfs" on that "cow" in AOTC, yikes), but other areas are quite well done. Yoda for the most part is tremendous, and I had a lot of people that talked to me and wouldn't believe me that the Clonetroopers were not real actors (especially in the close up shots). Heck, even the much-hated Jar Jar looked pretty realistic while walking in the sun and sands of Tatooine in Episode I. That said, the LOTR effects were just as amazing. I'd probably even agree to say that they were a little better. Amazing work with Gollum, and no real glaring problems IMO.
Without rambling on further, it is really hard to compare these two trilogies in particular. I have a tough time ever ranking anything above (or equal to) the OT, but the LOTR trilogy does come awfully close. In the end, it might be best for me to say it this way:
My Favorite Trilogy: Star Wars Original Trilogy
"Best" Trilogy: Lord of the Rings Trilogy
I really enjoy both of them, but Star Wars just has a special place with me. Lord of the Rings would be a close second, and I will admit, probably a better movie in the sense of acting, emotional involvement, and story. Star Wars is a more lighthearted and "fun" world to be in, and isn't necessarily about award winning acting (although the OT acting wasn't bad at all). The rest of the trilogies, to me, are ranked quite a ways behind these two. If ROTS turns out to be what it could be, the prequel trilogy for me will rank higher as well. Indiana Jones is not too far back either, and another movie trilogy that I find myself watching every time it is on TV it seems, at least for a little while. Great topic Brent, and a tough question to answer

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