I'm a fairly big fan of anime (and manga), but ironically, I don't watch a huge amount of anime. (I read a decent amount of manga in Shonen Jump, as I've noted in the past.) I'm also not familiar with a lot of "big name" anime, whatever that means.
My first experiences with anime were largely name-changing dubs such as Voltron and Teknoman (most people's above a certain age had the same experience, of course), but the first one that sticks out in my head was a series that was aired spottily many years ago on network television, Escaflowne (TVTropes page here) (also known as The Vision of Escaflowne and "Escaflowne of the Heavens"). The network in question, Fox(Kids), actually put together a pretty cool introductory segment, although they also rather mauled the series.
Not enough to make it unenjoyable, but definitely enough to "ruin" it, as they skipped the first episode and then stopped after a few episodes when they realized "Hey, this show has huge amounts of blood, violence, and other adult themes in it!" So you can't really blame them; the studio that dubbed it apparently wanted to put it on cable first. Eventually, it saw more proper releases, and it's apparently much more popular and well-known in western countries than in Japan.
Escaflowne is a really great series. It has giant robots (which I'm obviously a sucker for) in a medieval setting, and they're powered by magic rocks that are the internal organs of dragons. There are great characters (my favorite minor character is a guy who dies early in the series who jumped around only in normal armor while carrying a large sword and cutting arms off of giant robots) and strong and original writing. And it's even got a moving (if goofily shojo-esque) romantic plotline which is the crux of the series-which led to an episode where the villains plot to break the characters up because of how important it is. (Saying how and why would be huge spoilers. And it's much better than it sounds.) The primary villain of the series is also a historical figure, and there are a few really unexpected twists along the way.
It also has some incredible music.
There's also an alternate continuity movie and two or three manga adaptations out there, all of which take out crucial aspects of the story or otherwise make changes that mean I have no interest in any of them. (The movie is dumbed down to be a movie, and also makes things darker, edgier, and bloodier merely for the sake of doing so [and the anime was already pretty darned bloody, which means that it was pretty awful], although there was a cool fight between organic giant robots at the end. Each manga version took things in an opposite direction, one going more in the shonen direction and changing the characters to match [read: giving the boyish female protagonist more pronounced curves], and the other going in the shojo direction and removing all the violence, which means it probably was pretty boring.)
All in all, Escaflowne is probably the objectively best anime I've ever watched the whole way through.
My only real problem with it is that the characters are drawn with these crazy-looking big, pointy noses. That can be forgiven, though.