Here's an updated list of shit I recommend. I second the people in this thread that recommend Steins;Gate and Madoka. When it comes to Nasuverse stuff, everything ranges from watchable to good with the exception of the Fate/Stay Night anime adaptation. Avoid that at all costs, though I recommend the VN if you don't mind reading.
Accel World [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=13857] - A good series based on a futuristic video game called The Accelerated World. It's pretty stylish all round. (Spiritual sequel to Sword Art Online)
Denpa Teki na Kanojo [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=10519] - I made a long winded post here [http://www.escapistmagazine.com/forums/jump/18.831667.20306635] about why I think this is such a must-see anime.
Eve no Jikan Gekijouban [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=11350] - Set in a future where androids have become so indistinguishable from humans that they are forced to turn off all emotional programming and have a visible halo active above their heads at all times. There are however, places where androids are welcomed as they are with no restrictions and it's in these places that they can be who they truly are. But when their halo is invisible it's nearly impossible to tell if they're human or not. Time of Eve is the name of one such place where humans and androids can interact without ever knowing who is and isn't an android.
Mardock Scramble [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=11338] - Legion posted a good summary in this thread already.
Fate/Zero [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=12376] - This is the Nasuverse done right. Try your best to track down the BD subs for this if you can, it's simply stunning both visually and storywise.
Fate/kaleid liner Prisma Illya [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=15245] - If you aren't allergic to Magical Girl anime, this is a really fun show with lots of great action based on Nasuverse stuff. It's not canon though.
Kara no Kyoukai [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=7472] - A 7 part movie series with more excellent Nasuverse stuff.
Golden Boy [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=373] - No need to analyze this. It speaks for itself.
Hataraku! Maou-sama! [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=14846] An evil lord flees from his home dimension into ours after his humiliating defeat at the hands of a hero and must acclimate to Japanese society while trying to plot his return to power.
Isekai no Seikishi Monogatari [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=10393] - Tenchi Muyo spinoff that's set in another world as the title implies. Yeah, this is a harem but it's great when a harem lead is not afraid of his own cock, he just has better things to do. Kinda gotta watch it to understand.
Magi- The Labyrinth of Magic [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=14395] - Pretty cool story set in a middle ages Arabian Nights setting where magical labyrinths appear around the world full of treasures and Djinn that would choose the next king.
Mouretsu Kaizoku [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=11399] - Also known as Bodacious Space Pirates. Don't expect One Piece, it's a show about strong and cute female characters and politics and insurance fraud. And a whole lot of Space Ship porn on the side. The soundtrack is also absolutely lovely.
Nazo no Kanojo X [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=13899] - It's honestly as weird as it sounds but it's a lovely story that's one of my all time favorites.
Shiki [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=11302] - Slow starter but an excellent vampire horror story set in modern Japan.
Sakurasou no Pet na Kanojo [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=14238] - An excellent RomCom that's as good as if not better than ToraDora!
Sword Art Online [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=13858] - Set in the world's first Full Dive MMO environment where 10, 000 players are locked into the game and will die in real life if they die in game. Excellent first cour but the second cour lacks the tension of the first. Hope to see a season two for Gun Gale Online.
The Disappearance of Haruhi Suzumiya [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=11120] - Yeah, it sucks that the TV show has such an awful fanbase and the Endless Eight sucked too but this movie is an objective masterpiece. And I do not use that term lightly. Make Sure you watch at least episode 1-5 in chronological order, Bamboo Leaf Rhapsody and at least the first two and last episode of Endless Eight for maximum effect. This is a movie that actually managed to turn 5 minutes of deep introspection into one of the most tense and watchable scenes of all time.
The [email protected] [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=12436] - Based off an extremely popular Japanese video game series where you raise a group of 13 young, unknown girls to become top idols. Slow starter but it quickly picks up pace as the girls become more popular and it's worth watching for the performance scenes alone.
The [email protected]: Xenoglossia [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=7442] - Ignore the relations it holds to The
[email protected] and you have a giant robot anime that's so bad it's good. It's like watching a car crash in slow motion, it's so horrifying that you can't look away. The first episode sucks pretty bad but I couldn't stop watching after episode two.
Tasogare Otome X Amnesia [http://www.animenewsnetwork.com/encyclopedia/anime.php?id=13892] - This is a pretty cool and interesting specimen. It's more or less a horror/romance story for girls but is extremely enjoyable nonetheless.
Log Horizon [http://www.animenewsnetwork.com.au/encyclopedia/anime.php?id=15118] - It took a while for this show to grow on me. It's more or less about a bunch of MMO gamers that got sucked into the game and have to find their way in a world full of monsters and NPCs that have history and a will of their own. Small amounts of MMO and high amounts of politics and bureaucratic dealings.
Yahari Ore no Seishun Love Comedy wa Machigatteiru - My Teenage RomCom SNAFU [http://www.animenewsnetwork.com.au/encyclopedia/anime.php?id=14888] - Really enjoyed the cynical MC of this show because he reminds me so much of myself when I was around his age and thought I was some special snowflake. I very much liked the art and animation of this one as well. It's got RomCom in the name but the show is more or less a sarcastic deconstruction of the genre and stands out as having some of the best writing of 2013.
Unbreakable Machine-Doll [http://www.animenewsnetwork.com.au/encyclopedia/anime.php?id=15107] - This show has delivered unto us one of the greatest miracles of the Universe. And her name is Yaya. I want a Yaya. Also, the show has a pretty decent story and great action scenes.