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. thete1's "Flesh and Blood Like Anyone." Tim Drake stories drive a lot of this fandom, and this is not only possibly my favorite Tim story, but my favorite Te story ever. Te's one of my oldest friends in the fandom, but I'm not exactly her typical audience, because we like really, really different stuff. So when one of her stories hits my buttons, HARD, *look out.* This story is *extraordinarily* effective for me, and does a fabulous job of playing with the "multiple Robins" issue. Wonderful introduction to the multiple Robins, as well as being a terrific portrait of Tim. And understanding Tim will give you one character you have a handle on in the vast majority of fanfic, so it's easier to explore.
4. brown_betty's Heart, Humble. Tim Drake at another point in his career: *after* his Dad finds out he's Robin, and trying to deal with it. The story is his father's, rather than Tim's, but I think that could help a reader who's new to the fandom -- Jack has just found out that he doesn't know his son all that well, and because the story is about Jack coming to know him it can easily help a reader to do that too. Between this and Te's story above, you've got a really good picture of Tim Drake, and because Tim is so versatile, and useful in so many kinds of stories -- honestly, about the only major character he doesn't give you an angle on is Commissioner Gordon, and there's next to no fanfic about Jim anyway -- Tim stories are your way into the fandom.
5. astolat's "Reconcilable Differences," which is one of the best slash stories I've ever read. This is especially remarkable not only because I'm not a slasher, but because, although it interweaves with mainstream DC continuity, it's a SMALLVILLE fanfic. And I *loathe SMALLVILLE with every fiber of my being.* But astolat's love for the characters and the dynamic is extraordinarily winning, and the story just flows beautifully. In fanfic, it's easy to fall into the trap of over-writing, of wallowing in stuff, and there's none of that here. The story is very long, but it's all meat, and it just flows beautifully. Also, it's very funny.