FMA/Brotherhood
FLCL
Code Geass
Death Note
Ghost in the Shell
From the top of my head, of course. Would probably be a different list if I were to give it more thought, though those are most certainly some of my favorites, so whatever. (Monster, Zankyou no Terror, and Gurren Lagann all came to mind afterwards)
FMA Brotherhood
Mirai Nikki
xxxHOLiC
Tsubasa: Reservoir Chronicles
Black Butler I & II (Haven't indulged Book of Circus yet)
I have a lot of ones I like that I haven't finished/aren't finished (Steins;Gate, Fate/Stay Night: Unlimited Blade Works, Ansatsu Kyoushitsu, Blood+, etc....), but these are the ones off the top of my head.
These are just the five that come off the top of my head. At any one time, these can change, because there are a lot of good anime. Especially when you delete the entire Moe genre. I'm also sticking with TV shows, not movies. Maybe I'll do movies later if someone asks for it for whatever morbid reason.
#5: Hellsing. (The 2001 version.)
This guy is cool. No, seriously, he is. He's pretty much sealed badass in a can. This show screams "adrenaline pumping violence", and delivers on it beautifully with dark aesthetics. The dubs are also remarkable passable, but I wouldn't blame you if you switched it to subs. There isn't much of a plot beyond "go kill the thing Hellsing", and you know what? That's okay, not everything needs to be intelligent existentialism. If you ever needed the antithesis to Twilight, this is it: Gritty, violent, unashamed, teenage fetish fueled violence.
It's a delightfully odd science fiction anime featuring mechs with clear limitations and one of the best romantic arcs done on television (animated or not), with an interesting post-Earth dystopian universe. It's only let down by a bizarre ending where it wraps up its remaining plot threads in a very rushed manner. Still worth a watch though for some gory action. The dubs aren't too awful, but you probably wanna watch it with subs.
A bright and captivating science fantasy world featuring child characters that learn the harsh lessons of puberty and conflict. It also has a romance arc, but it's more aimed towards younger teens than young adults like Blue Gender is. You come for the cute, and stay for the emotional punches to the gut it delivers that are on par with this. Since that scene is considered one of the most gut wrenching in all of Western television, that sings high praises for this show's ability to twist with your emotions like a master puppeteer. I really can't say too much about this show without spoiling it either, it's full of plot. So go for it.
#2: Full Metal Alchemist. (Not Brotherhood, though that one is great too.)
Two brothers try to bring back their dead mother, causing one to lose an arm and a leg, and the other to be imbued into a suit of armour. It's graphic, violent, and painful. The heroes win some battles and lose others, with a surprising amount of moral ambiguity in featuring a lot of flawed people among the villains. The action is top notch and the journey makes sense. What ends up putting this up and above Eureka 7 for me is primarily the soundtrack. Without the show these touch your heart, with the show they take otherwise well animated and voice acted scenes and propel them into heart rending affairs.
Oh, and the dubs are top notch. Watch this dubbed.
The reason I prefer the original over brotherhood is spoiler territory so you've been effectively warned.
In brotherhood, the story is darker and tighter, there's less filler which is good. It remains closer to the manga, if that's important to you. However, the soundtrack in the original (in my opinion) is superior, it sticks with me. I honestly couldn't hum back Brotherhood's soundtrack to you if I tried, it's very generic orchestral. They traded the emotion for "moar noise", which suits the scenes at the time, but fades and ultimately perishes in the test of time in my mind.
Also, the original was more subtle with the flaws of the characters. Everyone gets amped up to 11 with their flaws, whereas in the original, Lust was a believable fatalist and you go from hating her to feeling sympathetic to her plight. Scar goes from scary malevolent force to a broken man who has lost his way and home. Edward Elric has moments where he almost breaks his own values, where he realizes that deep inside, he's fully willing to commit genocide if it means getting his hands on a Philosopher's Stone to restore his younger brother's body, and his younger brother has severe issues remembering his life and has to consider that he may not be real in the sense of possible merely being a byproduct of his brother's memories rather than the original person. All of this is told primarily as subtext: It doesn't deliberately beat you over the head with it, save when characters are lamenting their very human flaws. It feels more human, whereas in Brotherhood, they feel more fictional.
Either way, both series are great, and I could easily recommend either. However, personally, the original tops Brotherhood.
#1: Neon Genesis: Evangelion.
You can watch this show ten times and see something new and different every time. Every scene is chalked full of metaphors and underlying themes. This show is choking on symbolism so hard that Shinji would cream himself twice just knowing how mindfuckingly deep the rabbit hole goes. This is one of the most intelligent anime ever produced with top notch action sequences to go right along with it. It is, however, highly confusing, and makes absolutely no attempt to play itself to a younger audience, so it was whole heartedly made for adults.
As a disclaimer: The person who wrote this universe was in the middle of a deep depression and wrote it to try and give visualization to his emotional state. The entire show is the fantasy imagery of the mind of a suicidally depressed man. Watch with your symbolism glasses on, because every character is a piece of his fractured mind attempting to reconcile itself. Throw in heavy Biblical referencing and Christian iconography and you have an incredible complex and dense narrative. The fans of this show could go on for hours explaining it to you as well, so, if you need some help figuring it out, the fans are more than happy. Even the ending makes sense in this context.
The movies are also alright, but on a personal note, I find a lot of the more head trippy, bizarre shit has been toned down in favour of the violence and dumbing down for a greater audience. Still fun to watch, but I wouldn't put them in my top 5, as the show simply had more time to really delve into the deeper aspects of a fucked up mind.
Historical, fantasy, magic, horror, supernatural, survival, vampires, demons, pirates, mutants, ghosts, romance (FxF, MxM, MxF) (Romance should be part of the plot and not the whole plot in itself), etc.
Durarara!!
Attack on Titan
Ouran Highschool Host Club
Bleach
Neon Genesis Evangelion
I have a few others I rather enjoy, but these are the top five in no particular order. And no reason why I like them, too early in the morning for that kind of long post.
Fantasy is my #1; I will give almost anything a chance if it has strong fantasy elements. Post apocalyptic, superhero, alternate history, science fantasy, some supernatural, romance, and a few fandoms (especially Game of Thrones) are also likely to catch my eye.
Oh man this is hard ;~; I've not seen many though despite my list of 15+ to watch, so bear with me that this is a list from a very, very small pool of anime I've completed. I guarantee it will change, probably very soon too if I get some free time.
In no order...
Ouran Highschool Host Club
FMA
Code Geass
Blue Exorcist
Maid-sama