I was walking the streets of Madrid when I came across the national library. The only problem was, I wasn't allowed in the part where they had books. The building itself on the outside was amazing. I particularly loved the lights on both sides of the staircase that had lamp shades on them. Quite a nice touch.
Inside they have exhibitions and also the permanent collection, which is not of books as a library should be but of the history of books and printing and things book-related. It's a great place to visit and, best of all, it's gratis!