The border of Kentucky at what is known as "The New Madrid Bend" owes it's strangeness to the serpentine path of the Mississippi River. The River was supposed to define the western edge of the state of Kentucky and the Southern edge of Missouri. The result is an exclave peninsula of land in Kentucky that is completely surrounded by Missouri and Tennessee. To drive here from the rest of Kentucky, you must leave the state, drive through Tennessee and come back into Kentucky.
This spot along the Mississippi River is also the site of several historic events. Most notable is the "Battle of Island Ten" in the Civil War where Confederate Troops occupied Island Ten (now mostly part of the shoreline at the southern-most part of the bend in the map). The plan was to block and defeat the Union Troops coming down the Mississippi from the North at a slow point in the river. The plan didn't work so well. Although this was a potentially vulnerable spot in the river, Island Ten was also remote and could only be re-supplied by a single road through swamp and marsh land (the Union eventually succeeded in cutting off Confederate supply lines).
Mark Twain wrote about the area in his book, "Life on the Mississippi" and in particular about the feud between the Darnell and Watson families. One family was from Kentucky, the other from Tennessee. According to Twain, the two families attended the same church at what is/was known as Compromise Landing. The Church straddled the border between Kentucky and Tennessee enabling the families to walk up the aisle on their side of the church and attend services without stepping into the other state.
Lastly, New Madrid was the epicenter for several of the strongest Earthquakes ever felt in the United States in 1811 and 1812. According to Wikipedia, the earthquakes were so disruptive that they reportedly reversed the course of the Mississippi River around the area.