The old library at Trinity College. In here is the Book of Kells, a beautifully illuminated manuscript of the gospels which dates to about 800 AD. There's a very nice and extensive exhibit about the Book of Kells as well.
The tour also includes the Long Room, a very long, two-story tall room with the Library's rare book collection.
No pictures were allowed during the tour, but here's a Google Image search for the room, so you can get an idea of it. But pictures can't fully convey the size of the room. When I first stepped in the room I had to just stop and take it all in, probably with my mouth hanging open, for a minute or two before I could go on and actually look at the items on display.