The monastery with the chapel in a cave near Pythagorion.
Moni Spiliani is possibly the most visited monastery on Samos. I believe it is primarily due the its walking distance to Pythagorion because it is not the most charming or the most beautiful monastery on the island.
That said, it is still a pretty place with a small chapel in a cave behind the buildings. The cave is reportedly one of the mines where the building materials for the ancient city were quarried.