Rupert started it with this:
I answered by stretching his route slightly: Kantishna Station, Alaska to Skarsvag, Norway. It's a pretty long journey no matter how you look at it.
This one seemed like a good North American entry:
gets a score of 3.7.
But North America is tricky. Just about every goat and Jeep trail is mapped, and we Americans cannot abide straight lines that aren't paved. Rupert's still managed to find some good ones: Route to distance gives a very respectable 5.6.
1267 km by Google, 216 km straight arc. Score is 5.866.
It's kind of a pain because you have to snarf the lat/lon from Google's URL and adapt it to the great circle calculator, but it's fun to exploit holes in Google's map coverage.