1. Nodame Cantabile - just overall excellent, excellent series. The romance aspect is actually quite low-key (in comparison to some other examples I'm mentioning below), but it's no small feat to bring such outrageous characters together and make it both believable and rootable. Of course the story (and supporting cast) surrounding them is very interesting and hilarious on its own. Cannot be recommended enough.
2. Nana - since we're ranking anime (and not anime/manga), it's going to be second place. It's a tragedy that the series was never finished and although in the end it did become a little too melodramatic for my taste, it still handles its romantic subplots in an exquisite way. Sometimes it even hit a little too close home.
3. Paradise Kiss - honestly it would be no. 1, but while the manga is my favorite of all time, the anime just doesn't compare. It still has the best, most realistic characters and relationships I've ever seen in the medium.
4. Revolutionary Girl Utena - it does other things much better then the romance, this is why it's so low in my ranking. I still maintain that it's one of the best anime ever created, extremely smart and creative. (I want Bryan Fuller live-action adaptation)
5. Victorian Romance Emma - not much too say, but it's such a gorgeous, calming show. It has all the best elements of a period drama and may be one of my favorites in the genre (yes, even counting Western live action Austen adaptations)
Honorable mentions: Ouran High School Host Club (one of the funniest series ever), Lovely Complex, Toradora (the series surprised me with it's maturity, while still being adorable. Loved it), Skip Beat! (shame that the manga completely lost its plot, it is simply going on for too long - but when it was good, it was an excellent, addictive shoujo)