This was an interesting question. I thought about it for a while, trying to figure out what criteria to use. I finally settled on three things: 1) they help you figure out what the world is like and/or 2) give you a good idea of who the characters are and how they relate to each other and/or 3) make you realize how rich the setting is and the kind of things that are possible. Also, they can't be too "inside baseball."
1. "Regions of Sorrow," by E. Kelly. This is an older story, from pre-LJ explosion. "Regions of Sorrow" was posted in 2001, and it's one of the stories I read when I was just getting into the fandom as a reader. It has stuck with me ever since, and remains one of the very best pieces of fanfic -- not just Batman fanfic, but fanfic, period -- that I've ever read. It's not so much a Batman story as it is a story about Batman's world, and what it's like to live in Gotham City as just another person. It's convincing and utterly terrifying; the horrors take place off-stage and are all the more effective for it, because that helps to drive home what it must be like to be in Gotham City, where horror lurks around every corner and you never know when the next upheaval is coming. I reread "Regions of Sorrow" every so often because it works every time -- you can know what's coming and still feel the chills, because -- with little fuss, fanfare or atmospherics -- it's a quietly excellent story. And that's its magic.
2. Another older story, one I remember from getting into the fandom: Lys's The Beginning of a Beautiful Friendship. This is noteworthy because a lot of the fanfic I'd been reading was fairly hostile to Steph (again, this is five or so years ago, pre-LJ fandom explosion), and this rooftop meeting between Dick as Nightwing, and Steph as Spoiler, struck me for two reasons. First, because it was sympathetic Steph-perspective, which was unusual for the time... and second, because I really liked this Steph, a lot. It's a very good Steph introduction, I think, or it was for me, because this was the story that made me warm to her and think about writing her.
3.
4.
5.