I am, by and large, not a huge fan of anime. For the most part it seems to target a particular demographic of which I am not a part, and furthermore I find it annoying that many terrible shows will become popular for no other reason than being a part of that format. That said, there are certain anime shows from which I have derived great enjoyment.
Cowboy Bebop, for example, was a show that I absolutely loved. It had an interesting enough story to pull it together, but most of the episodes stood alone quite well. It's use of music is arguably amongst the best in television history, and its visual sense was far better than most anime I've seen. Far and away, though, the greatest thing about Cowboy Bebop was its characters. Sure, Ed was little more than comic relief designed to appeal to younger viewers. Past that, though, the leading characters were probably the best any anime has ever had. Jet's odd combination of hard-boiled film noir detective, bounty-hunting cyborg, and caring paternal figure made him a very unique character in the anime world both stylistically and behaviorally. Faye, who could have easily fallen into that "sexy dangerous woman with her own agenda" role that we've all seen 500 times too many is instead given an actual past, skill set, and collection of human flaws that make her deeper and far more interesting than most leading female action characters. And Spike, as corny as it may sound to say about a cartoon character named "Spike", is so compelling and fascinatingly complex that I think a good case could be made for including him on a short list of the greatest fictional characters of the 20th century, regardless of format.
A somewhat lesser know anime which I found highly worthy of praise was Ergo Proxy. That one is a tad tricky, though, since by its very nature it could never develop much of a following. It seems that the vast majority of rabid anime fans in the US are in the roughly 12-16 age range, give or take, and as is the case with most things teens flock to, most of the rest of us avoid it like the plague. The trouble with Ergo Proxy is that it's geared toward an adult audience. By that I don't mean that its content is especially lewd or graphic, though it does have a reasonable amount of violence. Rather, it should be watched by adults simply because I don't think most 14-year-olds would be able to understand it. It's too slow and too complex for younger viewers, who will almost invariably either get bored or confused or both. As such, I've noticed the show tends to have a sort of cult following, with those rare anime-watching adults who've seen it singing its praises, while the unwary teens who have picked it up all complain that it's "boring" or "confusing" or "not as good as [insert thing with lots of boobs and explosions here]" etc. Well for anybody over 18 who, like me, is still a big enough geek to occasionally dabble in anime: if you want proof that anime isn't just for pervs and weeaboos, I submit Ergo Proxy as Exhibit 1. And I say that realizing that I've given you no idea what the show is even about. I guess that's just a mystery you'll have to decide whether or not you care enough to solve.