The beautiful Monastery of Agios Panteleimona, located up a steep, winding road (not for weak of heart), is one of the most important sites in Tilos, and indeed the Dodecanese. Dating from the 15th century, it features a wonderful wooden iconostasis (1714) and religious paintings by Grigorios of Symi. There's a natural spring, beautiful cypress, plane and walnut trees, and stupendous views. For the walkers, a track leads down to Agios Antonios.
It's located at the literal end of the road, 14km northwest of Livadia. Check the hours before heading out as it closes for afternoon prayers; on Sundays in summer, the local bus runs here.