Presidential aspirant Tim Pawlenty (R) looks like he wants to extend his time as governor past his term of office, using a longshot dispute over the election for his successor as a lever to keep himself in office. Significantly, Pawlenty's post-governorship would be the first time he had a Republican legislature to work with. So it must be hard for him to resist that possibility with his run in the GOP primaries ahead of him. Interestingly, the apparent loser in the race, Republican Tom Emmer doesn't necessarily want to go along with it
To be clear, the Minnesota constitution calls for the departing governor to remain in office until the end of his term or until his successor is confirmed. So if the recount drags on, this is what the state constitution would dictate.
Remember, there was something quite similar in 2008 when the national GOP continued to fund Norm Coleman's court fight long past the time he had any chance of winning, because it kept Al Franken, a 60th vote from being seated.