If staying on the routes identified by the Mile Post, what is the minimal fuel range you would consider being required? And which route would be the worse case?
It varies a lot depending on what routes you're talking about. At least my edition of the Milepost highlights long stretches where fuel is unavailable in the descriptions.
On the Dalton highway going all the way up to Deadhorse, there's a stretch of about 240 miles without any fuel availability. The road is listed in the Milepost, but that doesn't necessarily mean that many people travel it for pleasure.