The Cave of the Sacred Icon of Agios Giannis features a large open cave plus white-painted rocks surrounding the tiny entrance to the main sequence of caves. A torch is useful for an initial hands-and-knees crawl along a low-roofed tunnel that leads to caves full of stalactites and stalagmites. The cave can be reached on foot from Agios Giorgios in less than two hours (one way) and in around 1½ hours via Panagia.
If you're on the island on August 28, it's well worth catching the Festival of St John the Baptist, held inside the larger cave.