I think this board needs a thread for recommending good series of fanfiction. A cursory search of the list of subject lines suggested that we have none so far, though I welcome any corrections. So I'm gonna start off this thread with two series I have grown fond of in the past six months.
First, there's "Harry Potter and the Methods of Rationality" by Less Wrong (and indeed inspired by the "Less Wrong" website). I loved it pretty much from the first chapter, though I am none too fond of the Harry Potter franchise as a whole. It can be found at http://m.fanfiction.net/s/5782108/1/Harry_Potter_and_the_Methods_of_Rationality in case you're interested.
The other series I would like to recommend is "Kyon: Big Damn Hero", a sort of crossover between the Haruhi Suzumiya series and TV Tropes. I would recommend reading the Haruhi novels before starting on this one. I had only read the first two novels before I started reading it, and most of the in-universe references went right over my head. Still, it is possible to read with nothing more than knowledge of the anime series, or so I am told. The best place to find this series is at http://pishoque.net/brian/fanfic/kbdh/index.htm
That's all I can recommend at this point. As a fan of the Star Trek tie-in novels, I would be especially delighted to find out about good Trek fanfic series I haven't discovered yet, but I've deliberately kept the scope of this thread as wide as possible in order to maximise participation. Bring it on, 4channellers!
Seconding HP&tMoR and K:BDH.
"Luminosity" + the sequel "Radiance" at http://luminous.elcenia.com/
I wouldn't touch the original Twilight series with a 10 foot pole, but the description of this alternative series intrigued me, and I like it.
its summary:
"Luminosity is an AU reworking of the Twilight story where Bella is a rational self-knowledge junkie. Inspired by 'Harry Potter and the Methods of Rationality'. Requires no knowledge of or affection for Twilight canon."
I'll probably post more suggestions later, I have read a ton of misc fandom fanfics: if I like a fic I end to read the author's other works, even if I know nothing about the source material (though I tend to read the wiki page about the source material to get the gist of it), and then the linked-to other people's fics that the author liked enough to favourite.