Unfortunately, I am not nearly as creative as some of the authors here that I so admire. I have acumulated a rather large selection of "favourite" fanfictions on my profile. They are all good stories in their own right. But some are better than others. Some, though I know what they are about when I read the title, I would not remember to list when asked for my favourite fanfiction. Below follow my eternal favourites that are not only the best pieces of fanfiction I have ever read, but at least among the best fictional stories I have ever read anywhere at all. The order is the order in which I have read them.
November's Hall of Fame (as of 30/12/2010)
The Lie I've Lived (by jbern)
The Amulet of Time (the first cyclus 1-4, as the second sadly never was continued; by Luna the Moonmonster)
The Brave New World (by bellerophon30)
Make a Wish (by Rorschach's Blot)
Lord of Caer Azkaban (again by Rorschach's Blot who must be among the most talented authors I know of)