The lighthouse at Fugui Cape (富貴角), the most northerly point on Taiwan. The government in Taipei does control territory further north, namely in the Matsu Islands. The weather here is often blustery but walking all the way around the cape - starting or ending in the quiet seaside village of Laomei (老梅) - is worthwhile.
Travel in Taiwan magazine recently visited this part of the world. Click here and flip to page 41 to see the article by Joe Henley, who's also written a Taiwan-themed thriller. For more about Taiwan's lighthouses, see this 2012 blog entry.