An "antilibrary" is simply all of the books you own that you have not yet read. In general the smarter a person is, the larger their antilibrary is.
This term was first popularized by philosopher Nassim Nicholas Taleb in his book The Black Swan. Taleb credited the original concept, if not the term itself, to famed novelist Umberto Eco, the owner of a large (30,000+ volume) personal library, who according to Taleb would divide people who visited and saw his library into those whose first question was how many of the books he had actually read, which was almost everyone, and those few who understood that books are not just for reading, that unread books are actually a far more valuable possession than read books (which after all, have already been read), and that a large library is not just an ego booster or a badge of honor, but an invaluable research tool.
According to Taleb, "Your library should contain as much of what you do not know as your financial means, mortgage rates, and the currently tight real-estate market allow you to put there. You will accumulate more knowledge and more books as you grow older, and the growing number of unread books on the shelves will look at you menacingly. Indeed, the more you know, the larger the rows of unread books."
Taleb proposes to call this growing hoard of menacingly unread books an "antilibrary." Taleb's point is that truly intelligent people will not take pleasure in what they know, or use it as an ego-booster, but constantly strive to learn what they don't know, and even more importantly, learn what they don't know in the second sense of finding out what kinds of knowledge they do not have. In other words, converting unknown unknowns into known unknowns. Ultimately, that is the power of the antilibrary - it converts unknown unknowns into known unknowns. Known unknowns have no power over you, because they can always be converted into known knowns if need be. It is rather the unknown unknowns, the things which are in neither your library nor your anti-library, that will ultimately be your downfall.