Saikano come to mind.
Why?
It takes the typical "magical girlfriend" story and completely and utterly deconstructs it in such a tragic, brutal way. Mix in a story of star-crossed lovers against a backdrop of war, human experimentation, psychological torment and eventual entire human elimination; you have a really interesting and horrific take on a boring romance story.
This story is brutal and intense ad not for the lighthearted.
If you want a happy story with a happy ending...Don't ever watch this anime/read the manga.
Paradise Kiss (manga is MUCH better, but the anime did a really good job nonetheless).
Paradise Kiss is one of the most realistic romance series ever. All the characters are individual, quirky, somewhat hard to like, but completely realistic.
It's a coming of age story that focuses on an 18 year old whose life is completely changed when she's offered the chance to become a model, and is taken away from her school-revolving world. Her eyes are really opened to the world from a different point of view and she meets a whole cast of interesting, highly colorful characters. She discovers and loses herself along the way and has a lot of "firsts".
The ending for this series is especially realistic and touching; and it is the ending that really makes the main character enter the real world. Depending on your point of view, this series either has a happy or a bittersweet ending. If you want a real-world, slice-of-life romance, check this one out.
(Also, check out any work by this manga-ka, especially Nana).
Jellyfish Princess (Kuragehime) is just SO unconventional and hilarious (and somewhat touching) that you just have to watch/read it.
Take an incredibly jellyfish-obsessed nerdy girl who is terrified of men, her nerdy social outcast friends (NEETs) and a cross-dressing, wealthy young man who enters their wold and you've got the most mental romantic comedy/finding yourself story.
Once night, the main girl character runs into an incredibly beautiful woman who eventually befriends her. However, this pretty woman turns out to be a young man who loves fashion and pretty dress, and who takes it upon himself to makeover the main character. Throw in the cross-dresser's 30 year old businesman virgin brother into the mix and there's your oddball romantic triangle.
The anime for this is pretty short but very fun; the manga's of course better and is still ongoing. Just like Paradise Kiss, this is a Josei series, so expect older characters and more realistic approaches to romance.
Kimi ga Nozomu Eien The Eternity You Desire (Rumbling Hearts)
This is a highly controversial, and bittersweet/tragic romance. Based on a visual game, the anime is a short, but very realistic (almost painfull so) series about a young couple whose life is changes forever when the girl is involved in a horrible accident and falls into a coma for years.
(The first two episodes of the series focus on the start of the relationship, and ends with the accident). The rest of the episodes deal with the aftermath of the accident, the coma, and the girl waking up).
Then she wakes up...To discover that everything has changed for the worse, and everyone she knew -including her beloved boyfriend- has moved on and left her behind. Everyone lies to her about what has really been happening, but she eventually realizes how much time has passed and what has really happened. She tries to fix things and regain her place with her boyfriend but she realized just how hard that is when she doesn't know the people her friends and family have grown into.
This is a pretty sad show that involves betrayal, lies, and cheating. But depending how you view it, you might think differently. Again, this is a mature show so...
In fact, the ending for this series sparked so much hate, that an OVA was created with a totally different turn out of events and a modified storyline with a much happier ending (for those that wanted this ending).
Other series I would HIGHLY recommend and which are NOT your regular, boring romance series:
Victorian Romance Emma (realistic period romance; the manga is just amazing).
Honey and Clover (slice of life at college; very funny and very bittersweet).
From Far Away (one of the BEST fantasy mangas ever; a lot of non-annoying/cheesy romance as well and an awesome storyline)
Saiunkoku Monogatari (reverse harem that's REALLY well done, with lots of fantasy, mystery, romance, adventure...)
Anything by Clamp, especially their older series like
Cardcaptor Sakura, Tokyo Babylon and it's somewhat-sequel
X.... However, Clamp's mangas feature ALL kinds of romance -het, yaoi, yuri etc., but nothing ever explicit ever; not even kissing).