Dublin Castle, Dublin
Favorite thing: If you're not familiar with Dublin, you might look at this picture and think it's out in the country at some distant castle. However, Dublin Castle is actually right in the middle of one of the busiest parts of the city and is a nice diversion from the busy city streets.
Favorite thing: visit Dublin Castle. When I first saw it I was disappointed in its un-castle-like aspects, but the interior is beautiful. After touring, I looked again at the outside, and found it really does retain some castle-like qualities.
The Chester Beatty Library, Dublin Castle. The library opened in the castle in 2000 and is a fascinating collection of books spanning nearly 2000 years - from papyri to beautifully bound bibles. Having an interest in bookbinding and lettering, I could have spent hours here and plan to visit again. Some of the books were beautifully illuminated and richly decorated.
Absolutely everything you need to know about the Library is located here: http://www.cbl.ie/home/