I have three recommendations for you, DW. I'm going to suggest mostly serious stuff, besides my third suggestion. The humorous anime I've watched are more like FLCL. So I'll leave those out for now.
[b]Ergo Proxy[/b] - I don't want to explain too much about this one, since it will ruin the surprise, but it's a sci-fi anime that takes place in a future utopia. Humans have androids living in their society that are called autoreivs and they all live in and are protected from the outside world by a giant dome. It's pretty well done and I have few complaints about the series so far. The superpowers later on are a little cliché but most usually are.
I haven't finished it yet, but what I've seen I like so far. No anime sweat drops or anything and the characters are more like real people. Also, it has the most action out of the three I'm recommending.
[b]Dennou Coil[/b] - A neat little anime about the supernatural and virtual reality as seen through glasses. The kids (yes, it revolves around kids, so that might be one reason you won't want to pick it up,) have anime eyes but it usually ends there. Someone's aunt though is actually a high school student, but she seems like an adult. That's the only thing I can think of for a downside.
Besides that, it has a good story, some humor, a lot of mystery and it makes you feel for the characters. I finished it and liked it.
[b]Welcome to the NHK[/b] - I'm just throwing this one out here because I liked it so much. XD It deals with what you hate but it realizes how bad it all is. (The cutesy anime girls, etc.) it's about a hikkimori shut in who is pretty paranoid and believes that the NHK is making everyone into hikkimoris with anime. He ends up trying to go outside more and meets an old school friend who wants to design perverted dating sims. He also meets a girl who wants to help him escape being a hikkimori.
Like I said, it deals with the subjects you don't like in anime, but besides in one episode or two, it never does anything but show them as anime figurines and stuff. The downsides are, the main character and his friend are perverts and it does drag on, but it has some good funny moments and good drama. It also makes the characters seem like real people with real problems. For example, it never tries to make the main character seem like a great guy. He's just who he is.
Well, sorry for the long read, but I had a lot to say. d: I hope out of one of these (Ergo Proxy probably) you'll find something you like.