Might be a strange recommendation, since it's a romantic comedy, but I recommend checking out Toradora. It does a good job capturing Japanese high school life, and so you'll feel a bit more immersed in it than with long-running battle anime. I know a lot of people that claim it was their gateway into watching more slice-of-life or romance anime, so I'd definitely try it out if you want to broaden your horizons.

If you're looking for something recent in the same vein as what you've already watched, I'd give One Punch Man, My Hero Academia, or Hunter x Hunter a try. Jojo's Bizarre Adventure is also worth a look-see, though it might be a turn-off if you don't watch much.

If you'd like to try something different, that you can enjoy regardless of being an anime fan or not, I'd recommend 91 Days, Erased, or Psycho Pass. 

Feel free to stop by the Anime Thread if you want to see what's popular, or get a feel for what might be interesting to you. The polls are sometimes terrible, but overall there's a decent community here.

I think a lot of good shows (and a few not so great shows) were already recommended here, so I wont bother to add to that list. That said, I think a hugely important skill for an anime fan is the ability to find good shows (or rather shows that they will enjoy). When I started watching anime, I watched some amazing shows and some crappy shows but all of it helped me develop my tastes and learn how to find anime that would appeal to me.

Wtf are people doing not recommending Death Note?

Countless of people(including me) gave anime a chance just because of it. Easily top 10 anime.

Death Note
Berserk
Cowboy Bebop

Great choice, the creator of that one also made Cowboy Bebop. Which is also legendary.

I just started watching monster recently. As of now, it's my fourth(or fifth- would you consider Naruto and naruto shippuden different series?) anime series. Finished 1 season of black butler, completed death note, Attack on Titan, Naruto, and am currently at episode 300ish for Shippuden.

I'm not an anime expert, but I realized that I like the pacing and other things in Eastern shows much more than shows in the US/Europe. Maybe it's because I've watched countless of superhero, alien, and detective movies. Either way though, I usually find myself enjoying shows made in the Middle East/Asia more than ones made in the west. Not just anime. Even dramas manage to be superior to the ones we get in the west.

I know this is about anime, but if you want to watch a show that isn't anime, I'd recommend you look at Arabic shows. 

Bab Al-hara- A show about a syrian village ruled by French authorities: http://www.imdb.com/title/tt1999065/

Omar- a show about one of the most important leaders of Islam: http://www.imdb.com/title/tt2063241/?ref_=tt_rec_tti

As for anime, I know a lot of people would say my taste is shit for liking Naruto, but I never understood the hate behind it. Especially considering that it's Shonen, it's far from the worst. So yeah, I'm sure you probably just laughed and skipped everything once you realized that I thought Naruto was great.

I'm fine with people recommending non-action anime. I just wanted to give people some info so that they would have an idea of what to recommend. I'll check that anime out if I have the chance. Thank you!

One Punch Man is possibly the worst one you could watch as an anime neophyte.

It's a parody anime. You'll get the most out it after having watched other shows first.

So One Punch Man is like Superman? Or like how in Persona 4 Chie does her Galactic Punt?

I don't hate NGE. It's a true masterpiece. Not in the fan boy of saying that anything they like is a masterpiece, but in the sense that is so good, well written and beautifully drawed that it transcends it's medium of Anime and turns into a piece of art. Shinji is easily one of the most well written characters in any media, he could be read as just a whiny boy, but he is much deeper than that. But it's depressing. Very depressing. Mainly The End of Evangelion. 

But to OP,it's more like of a deconstruction of the Mecha genre, so it won't make as much sense until you have watched other more typical mecha series like Mobile Suit Gundam.