There's a small isolated piece of Kentucky that isn't actually connected to the rest of the state ("exclaves", geographers call such things) known as "Kentucky Bend", or more colorfully, "Bubbleland". The exclave was created due to the shift in course of the Mississippi after the New Madrid Earthquakes in 1811 and 1812.
Located in the southwestern corner of the state, Bubbleland is a curiosity in many ways. Although the area is 17.5 square miles, only seventeen people live there according to the 2000 census. And despite being declared part of Kentucky after a long and protracted squabble with Tennessee over the land (who actually owned it as recently as 1848), the official post office mailing address for the area is nearby Tiptonville, Tennessee. (Since rockabilly legend Carl Perkins is from Tiptonville, this could in a sense make Perkins an unofficial Kentuckian!)
Mark Twain, in Life on the Mississippi, wrote about the feud between the Darnell and Watson families in Bubbleland, one that lasted six decades!