The largest swing I could find towards Barack Obama in the contiguous states was in St. Bernard Parish, Louisiana, southeast of New Orleans, where he got 36% of the vote against Mitt Romney, a sizable improvement over his 26% against John McCain in 2008.  However, I came across some Alaska election data today, and discovered some even more astonishing, massive improvements for President Obama from 2008 to 2012.  Take a look at these maps:

(Note: these are results from Alaska's state house districts, not it's counties (boroughs).  I could not find borough results anywhere.  The state house districts are not exactly the same shapes from 2008 to 2012, as there was redistricting, but you can pretty much see what was going on....)