La Fortaleza, also called the Palacio de Santa Catalina, is the official residence of the Governor of Puerto Rico. The original sixteenth-century fortress has been swallowed by four centuries of additions; the pink-painted neoclassical front you see from Calle Fortaleza dates mostly from the 1840s, but two of the original towers are still standing.
The building is a working government house, but the public rooms, the Moorish gardens, and the chapel can be visited on a free guided tour (in English or Spanish) several times a day, Monday to Friday. You need ID at the gate and dress is enforced — no shorts above the knee, no flip-flops. Tours fill up, so arrive early in the morning.