Looks like we've probably got the answer for all time and since the Civil War.
Since the passage of the Voting Rights Act in 1965, I think Obama's 92.46% in DC in 2008 is the highest for any subdivision, and his 71.85% in Hawaii in 2008 is the highest for any state for a Democrat. The best I can find for Republicans is 78.2% in MS in 1972. I think Romney's 72.8% in UT is second-place for Republicans in a state, though Trump won NE-3 with 73.9%.
Nixon got 75.04% in Georgia and 73.70% in Oklahoma, and Reagan got 74.50% in Utah, so Romney's 2012 performance in Utah would be in
fifth place for Republicans in a state.