The Island of Lipsi
Tourists seem stuck on a broken record of high-profile Greek islands. But for your own private beach and a family that welcomes you as a guest rather than as part of the economy, choose an island that has no promotional budget. Hop off the boat at an island with just enough commerce to keep you fed and watered.
Catch the daily boat from Patmos or Samos to Lipsi. The few families who rent rooms will greet you at the port with invitations to follow them home. While lunch cooks, practice your Greek with new friends over backgammon on the town square.
For a smile and a few euros, ride a Lipsi taxi (a pickup truck with a bench out back) to your own private beach. Set lazy afternoons free as wavelets tickle your toes and skin that's never seen the sun gets a private initiation.