The winner is 91.9/680 KBRW in Barrow Alaska. This distant outpost began broadcasting on December 22, 1974. For an radius of over 150 miles KBRW is the only radio station, that makes them the only radio service for every community in an area of about 88,000 square miles! Most of that is tundra but whatever. The nearest full service FM station is 88.1 KCDS a 90 watt community station over 200 miles away in Deadhorse, AK. "Remote" is an understatement. See pic here.
It was started with a $180,000 grant from the State of Alaska. About two years later KBRW began broadcasting with 1,000 watts of power and a small army of community volunteers. in 1988 they added a set of 5 translators to improve coverage in out-lying villages.
Barrow, Alaska is the United States' northermost town, situated 330 miles north of the Arctic Circle, a mere 1200 miles south of the North Pole. The Chukchi Sea in the Arctic Ocean surrounds Barrow to the west, north, and east. To the south,permafrost stretches 200 miles, all the way to the Brooks Mountain Range. The municipality of Barrow was named after polar expedition sponsor Sir John Barrow in 1826. Previously it was known as Utkikvik, which translated from inuit to "place where the snowy owl flies." More here.
If it interests you, the northermost radio station in the contiguous 48 states is a tad harder to discern. The lands along 49 degrees latitude are largely unpopulated, much of it is national forests. But in a few small towns there are local radio stations.