Some great ones already mentioned so I'll offer my own:

Midori Days (a guy wanting a GF really badly ends up with his right hand turned into a girl who's been obsessed with him from afar, quite funny)
Stein's;Gate (unique kind of time travel story involving a microwave, texting & green jell-o, more to it but also weird)
Bakemonogatari (a guy helping out cute girls he meets with hidden supernatural problems)
Gurren Lagann (While drilling the main character Simon discovers a mecha with special ability to combine with other mechas)
Red Garden (four girls become chosen to do battle with men who turn into wild doglike creatures at night, different animation style than usually seen)
Code:Geass (like Full metal panic but more complex story & characters), Noticed this was already mentioned above.
Monster (Slow & kind of like Fugitive with Harrison Ford)
Last Exile (Steampunk airships following brother & sister delivery pilots who are a lil too upbeat for all the death & destruction happening around them)

Anime Movies:
Howl's Moving Castle (Surprised nobody mentioned this classic!)
Summerwars (Guy visits his girlfriends extended family when the internet gets taken over and it becomes up to them to fix)
The Girl who leapt through time (pretty self explanatory)

Live Action South Korean Drama's: (for the times you get sick of anime but still want to read subs on everything)
City Hunter (Best one, lots of action & story. Child trained in the art of revenge/humiliation to rectify a top rank cover up years before claiming the life of his fathers platoon)
Dr Jinn (Modern Doctor sent back to Feudal Age from a tiny baby found in a trauma patients brain)
Rooftop Prince (Feudal Age Prince & bodyguards sent to present day)
Personal Taste (Lee Min-Ho pretends to be gay to copy an old-styled house to win a building contract, while engaged to another female & befriends the female owner, the daughter of the man who designed the house for his late wife, funnier than it sounds )
Secret Garden (Accidental male/female body switching, she does stunt-work he's a claustrophobic rich boy)
Big (50% chance you either like or hate this one)