Tinos (Τήνος)
An island unlike any other. Not much in the way of traditional holiday tourism, though that is changing. Lots of Greeks come here on a pilgrimage to see the miraculous icon at Panagia Evangelistria Church, one of the holiest sites in Greece. Quite a few ex-pat French now live on the island as well.
It is worth spending a few days to explore the island as it has some fascinating villages and some good, well waymarked, walking routes.