There are quite a number of unique books in my library, for a variety of reasons. In fact, as of this morning, it appears I have exactly 100 uniquely held volumes. Most of them are Jewish religious esoterica or Jewish children's books. It's possible a few of these are held by others and need proper disambiguation, a project I should take on one of these days. A couple are books on specific special needs, published by support associations or family/patient support groups; it's not surprising that there aren't other people on LibraryThing who haven't cataloged, "The Terrific Tale of Teddy the (Previously Troubled) Tummy Who is Really Happy Now and Lovin' Life: A Book to Help Kids by APFED." I really need to upload that cover, though; Teddy's cute. Maybe I need to somehow attach a photo of Sachy with his plush Teddy the Tummy, too. Several are needlework or costume books which I collect as avidly as I collect smocked and embroidered baby wear; these include obscure volumes like "Great Costumes 1550 - 1950" which is an annotated museum catalog. I've had to hand enter all of these; there have been a number of works I didn't expect to find already in the system, but which it turned out one or two people did already own. Those are almost all Jewish or children's books.
What my kids consider the most unique volume in our collection, at least that I've catalogued so far, is our edition of "The Poetical Works of Henry Wadsworth Longfellow: Household Edition" for while many do own copies of Longfellow's complete works, our volume was printed in 1881, and is still in quite fine shape.
There are about 32 other users in my "You and none other" feature list, but several of those other members share multiple unique listings with me. These are primarily again either religious books or obscure but not rare cookbooks.
I have found a few friends through the "You and none other" feature, and through matching unusual volumes in our library. I know that if others have several children's books meant for Orthodox Jewish families, they're likely to be Jewish and have children; if they have a number of odd but not rare cookbooks, they're probably like me prone to cookbook obsession. If they have books on now-much-neglected needle arts like smocking or French heirloom hand sewing, they just might be fabric or fiber obsessed, poor things. I do like friending people in LT, even though I don't in most other sites; here on LT I really feel like I have a connection with people who go to the trouble of acquiring and cataloguing books similar to those I treasure.