Golly gosh I dunno if I have one of these! I'm so bad about reading books even twice! I'm very much a "get it done and move on" kind of reader, which is lame and I really want to change it. I love it when people talk about books they read yearly, or books they love so much that they just read them again and again, and I want to experience that. I don't have that one book that is all battered and loved from years of rereading. Which makes me sad.
Here are some books I could imagine developing such a relationship with, some of which I'm sure you'll hear about later: