It’s funny that you complain about anime by saying:
*Cowboy Bebop
Ghost in the Shell: Stand Alone Complex
Black Lagoon
Neon Genesis Evangelion
Jin-Roh
*
…just for starters.
It was Ghost in the Shell.
FWIW, I hated Ghost in the Shell; after seeing it, I was convinced it was anime in general that I hated (I’d seen several before and not much cared for them; folks assured me that GitS was different, but it lived up to its acronym).
Princess Mononoke convinced me not to judge all anime the same. I was really surprised: I had to be dragged to the movie, and expected to hate it.
Daniel
Ah, okay. I assume you’re referring to the first GitS movie (1995), and not Ghost in the Shell: Innocence (2004) or the TV series, Ghost in the Shell: Stand Alone Complex.
Assuming that’s the case…yeah, actually, I didn’t care for it either. Mamoru Oshii really gets into reflecting on the nature of the self, which I can appreciate, but it makes for a rather turgid film.
Innocence is rather more straightforward, and Stand Alone Complex is pretty awesome TV.
Also rather more depressing
Plus it has the Tachikomas. I don’t think it’s possible to not like the Tachikomas. Watch it watch it WATCH IT! Yeah, I’m a fangirl, why do you ask? 
I watched a couple of episodes of GitS: SAC and it seemed decent enough but didn’t really hook me. The opening songis unforgettable though:
Naladubuites’!
Naladubuites’!
Aeria Gloris!
Aeria Gloris!
One of my Roomates doesn’t like the Tachikomas. Of course, he’s seen maybe 5 random minutes of the entire series, when he’s walking in and out of the room, and claims not to like anime at all(though has a DVD of Full Metal Alchemist that he won’t admit to having in his desk). So, yeah.
Speaking of which, and possibly derailing the thread, what is that little song they sing at the end of 2nd Gig?
Berserk. Well, it has plenty of violence, anyway. And some attempted rape.
And plenty of Nightmare Fuel at the end.
I’m not a big fan of anime. I’ve got a huge DVD collection of Asian movies (400 or so), and have probably seen at least that many on rentals. Mostly Korean and Chinese, but well over 100 Japanese as well. And I love U.S. animation, but just not anime that much.
I tried Cowboy Bebop. I’ve seen **Akira **and **Ghost in the Shell **and **Paprika **(didn’t care for the movie that much but LOVE the music and bought the CD). Akira and Ghost in the Shell totally lost me.
I saw **Princess Monaoke **at the theater and didn’t really care for it.
I saw **Spirited Away **at the theater and thought it was pretty good. No great shakes (couldn’t see what all the hype was in the review I read that prompted me to see it). But it was enjoyable enough.
HOWEVER. I watched Spirited Away again on a rental, this time using the Japanese subs, and it totally blew me away! I couldn’t beleive it was the same film I had “meh’d” over previously. There was just so much more depth the second time around. I credit most of that to watching it in the Japanese dialogue, although I admit some of it may just be the benefit of a second viewing.
It has since become one of my all-tme most beloved films.
But I still don’t really like or “get” anime. And yes, I did try my re-watching experiment on Princess Monaoke but still thought it was not so good.
I know you said you fell asleep during it, but I would heartily second you trying Princess Mononoke again. I loved Nausicaa, but for me, Mononoke stands head and shoulders above everything else he’s done. It’s the pre-eminent big-screen fantasy epic, in my mind, dwarfing even the Lord of the Rings film trilogy. I know it’s a matter of taste, but the way Miyazaki presents the various interests in the conflict, how they ally and collide, how the themes of nature and industry and brotherhood and love play out in subtle, sophisticated ways- it’s unparalleled among fantasy epics and quite rare in film, period.
As for the OP, I liked Spirited Away, but I wasn’t blown away by it. It was cute, magical, and mysterious, but it seemed to lack the weight of some of Miyazaki’s other works. Not that every film needs to be weighty, but the ones that pull it off well tend to make a deeper impression on me.
Princess Mononoke is my favorite too and in fact one of my two or three favorite films of any kind. I believe it was the first anime film I ever watched and even though it was a pan-and-scan VHS tape with the dub I was completely hooked.
Miyazaki films, including Spirited Away, sometimes fizzle out at the end but Mononoke is an exception and the climax is truly awesome in the original sense of the word. Lady Eboshi is one of my favorite movie characters; I would love to see a film or series about her backstory and how she became leader of Iron Town.