Suggest a single anime? Hah. Have a dozen or more instead.
If you haven't seen any of the following list, some of the more well known and popular anime among English speaking folks, they're worth watching for a variety of reasons: Fullmetal Alchemist, Fullmetal Alchemist: Brotherhood, Neon Genesis Evangelion, FLCL, Cowboy Bebop, Samurai Champloo, and others of that vein. Y'know, the ones that aired on Adult Swim and Toonami and such and got a lot of people into anime to begin with. And now for ones I'll give specific reasoning for.
If you're interested in a short supernatural horror/mystery, Another is a decent one. Kind of slow to build, but it's worth it. There are some crazy death things but it doesn't devolve into torture porn like that one Nat suggested.
Tengen Toppa Gurren Lagann is basically Gainax's take on a mecha anime. The whole thing is essentially one giant "let's make shit bigger and cooler" build up until you get to the point where they're fighting in a mecha that uses galaxies like fucking shurikens. It's also got some pretty good character development going on, which is a big plus.
Code Geass is an alternate history/future anime with mecha and some supernatural shit going on. This is one of my favorites because it's full of shenanigans. The plot is that Japan was taken over by Britannia, their alternate history American empire, and they use mecha and stuff to oppress the Japanese people; then in comes the main character and rebellion happens. The main character, Lelouch, is a showboating giant ham who pulls off all sorts of shit just because he's ballsy and smart enough to make it work. There are also some other rather flamboyant characters who make it a fun watch, but Lelouch is the main draw.
Bakemonogatari is.. how the hell do I describe it? Uh. Sort of a mix of tons of different supernatural stuff, I guess. The main character has had quite a bit of experience with the supernatural and does a little work as a sort of occult detective, assisting a guy who's an expert in it. Mostly he happens to stumble upon people with supernatural problems and goes to the guy to learn how to get help and/or learn how to fix it. There are some major romantic elements to the plot as well. Also some interesting animation and choices with how they show things and give you information. It's one of those things that I think you kind of have to watch for yourself to understand, because a textual explanation falls flat.
Then, hmm... Oh, Mahou Shoujo Madoka Magica. It looks like it's going to be a bland magical girl anime for children, but then woahshit grimdark drama and horror. For the first two episodes you might think you're wasting your time with some cutesy anime (unless you pick up on the hints at seriousness to come), but then episode three comes and turns it all around.
If you're into heavy fanservice wrapped around a moderately interesting plot, Queen's Blade might be right up your alley.
Mirai Nikki is good if you just want some mindless entertainment and don't intend to get too analytical with it. Lots of crazy shenanigans happen in this one, but the plot is wonky.
Kore wa Zombie Desu ka? is good if you want something fun and entertaining that doesn't take itself seriously, a comedy with all sorts of fantasy and supernatural stuff going on. For example, the main character is a zombie who also becomes a magical girl because reasons and has to fight while wearing a pretty pink dress and wielding a chainsaw.
That's all I can think of off the top of my head.
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Space Battleship YAMATO 2199.
It's an ongoing reboot of a classic 1974 series which was beloved as part of the start of the anime revolution. Plus it's really for good sci-fi space battles if you're into that sort of thing.
Currently working on my first Advanced RP! Please bear with me, for I have not done an ADVRP before. I am open to constructive criticism!
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I'd watch Fairy Tale more, but honestly, I don't have the patience to invest in a One Peice level anime with a billion fucking years worth of episodes.
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Is that last one the one where he fights a giant fucking talking lobster in a tuxedo in one episode? I saw that clip once and was like, "Dafuq?"
“Do not worry about your life, what you will eat or drink; or about your body, what you will wear. Is not life more than food, and the body more than clothes? Look at the birds of the air; they do not sow or reap or store away in barns, and yet your heavenly Father feeds them. Are you not much more valuable than they? Can any one of you by worrying add a single hour to your life?"
"The broad wall of Babylon shall be leveled to the ground, and her high gates shall be burned with fire. The peoples labor for nothing, and the nations weary themselves only for fire."
"It is more blessed to give than to receive."
"For the word of God is living, and powerful, and sharper than any two-edged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and is a discerner of the thoughts and intents of the heart."
"Though an army besiege me, my heart will not fear; though war break out against me, even then I will be confident."
"Let us hold fast to the profession of our faith without wavering (for He is faithful who promised), and let us consider one another to provoke unto love and to good works."