If we’re talking about a Super Bowl winning year I’d have to go with 2001 simply because it was the first time and they overcame so much to finally raise the Lombardi. The other Super Bowl wins they were far more established and talented, almost to the point where if they didn’t win it, it would’ve been a disappointment.
As for a non-Super Bowl winning year I’m going to go with 2013. Just absolutely devastating injuries coupled with near complete turnover at wide receiver and yet they still somehow clawed their way into AFC Championship. I’ve never been more okay with a playoff loss in the Pats’ run. Also you have to mention the incredible late comebacks they had that were just so exciting.
I’m sure 2007 deserves some mention but that was in essence a pissed off all star team. Personally I enjoy seasons where major adversity is overcome as a sign of a great team.