Changdeok-gung (Palace of Illustrious Virtue)
Changdeok-gung was built from 1405-1412, burned to the ground by the Japanese, and rebuilt in 1610. It was meant to be an annex to Gyeongbok-gung but ended up being used as the seat of the gov't and royal house until the 1870s.
It's a UNESCO World Cultural Heritage Site and smack dab in Seoul. When you're inside, you often can't see any of Seoul's buildings and it's very tranquil.