Kalymonos or Kalimonos, not sure if it's a Y or I as I've seen both spellings. Another Dodecanese island west of Bodrum and lies between Leros and Kos. It's population is around 17000 and it appears to be one of the wealthiest islands in Greece!
With its own unique shops, boutiques and cafe bars it certainly looks like it. You won't find any Versace here (thankfully). No high street stores either, just it's own local brands and designs.
A little historical fact which lends itself during the Italian and Turkish war in 1912 it was once occupied by Italian sailors. Italy took control of the island until 1947.
Sponge diving was (and still is) another source of income. 
