don't have a ton of input as far as across the country. but my definite favorite is Cassidy bridge campground, SC. rent the entire campground for $30 a night, and down the mountain trail behind a locked gate, no one to bother you and its the most peaceful place on the planet IMO. In the county around, there are 42 waterfalls, endless miles of trails including the GA traverse and so much more.
if primitive isn't your style(no electric or water), Devils fork campground on lake jocassee isn't far and its pretty amazing if you can get a spot.