One Piece: An epic story about a 17-year-old kid who looks and acts 10, who is on a pirate journey to find a treasure called the One Piece and become King of the Pirates! He ate a devil fruit, though, making him unable to swim, but he is still ambitious about it! The devil fruit allowed him to stretch and stuff like that. Sounds stupid, but it's actually pretty awesome. The anime also addresses real world issues such as bigotry, depression, censorship, etc.. It's long, but worth the watch.
FMA :What Lios said.
Death Note: There was once, an innocent and intelligent student. And then he found a notebook called the Death Note. Will he use his notebook for good, or will he abuse his power? This anime can also be an allegory for either the Bible or politics.
The Melancholy of Haruhi Suzumiya: Although Melancholy is in the title, it's not even a sad anime. Idk why it's called that, tbh. But anyway, a normal guy is just having his normal day until a hot girl transfers and seeks aliens, time travelers, and espers, which the main character thought to have not existed. After the hot girl forces the main character in and recruits some other members, everything changes. Now they go on adventures to solve mysteries and stuff like that. It's a fun anime to watch. There's a specific chronological order to watch it in, though, not all websites have it in order. So look up what order to watch it in. And when you get to season 2, only watch the first, second, and last episodes of the Endless Eight arc. Between that, it's the same thing over and over again.
Baccano!: It's sorta confusing since they jump between time periods every episode, but it's still pretty cool. It's basically about immortals and stuff like that. It's pretty comical, too.
Durarara!!: There are a bunch of street gangs and a headless rider is looking for her head, IDK HOW TO EXPLAIN ALL THIS. XD
Steins;Gate: Well, a man, who is a scientist, figured out something called a D-Mail, allowing people to sent texts to the past. However, one simple, harmless text could really mess up the future.