I'm not a big anime fan, but there are a few I really like.
The one that really got me into the medium was Serial Experiments Lain, which defies coherent explanation. It's a fun cyberpunk SF series that ends up delving into some surprising areas.
The character designer for Lain went on to make a really strange anime about a poor student and an alien called Niea Under 7, which is also fun to watch. The thing that both of these shows have in common (and it's something that I haven't really seen in most anime) is a deep exploration of what it means to be connected to other people.
A third that I really enjoyed was Bubblegum Crisis, both in it's original form and the later reboot. The reboot looks better, but the original series definitely went a lot further in both it's exploration of what it means to be alive as well as how well realized the characters are. Both definitely delve into that same field: what does it mean to be connected to other people, so I guess you can see a trend here. Anyways, I highly recommend these.
The one that really got me into the medium was Serial Experiments Lain, which defies coherent explanation. It's a fun cyberpunk SF series that ends up delving into some surprising areas.
The character designer for Lain went on to make a really strange anime about a poor student and an alien called Niea Under 7, which is also fun to watch. The thing that both of these shows have in common (and it's something that I haven't really seen in most anime) is a deep exploration of what it means to be connected to other people.
A third that I really enjoyed was Bubblegum Crisis, both in it's original form and the later reboot. The reboot looks better, but the original series definitely went a lot further in both it's exploration of what it means to be alive as well as how well realized the characters are. Both definitely delve into that same field: what does it mean to be connected to other people, so I guess you can see a trend here. Anyways, I highly recommend these.