Upgrade of a many TB postgres database with little down time

Big (as in many TB) databases have complex management problems. One of such problems is upgrading, a process that normally can imply even hours of down time.
But not all businesses can afford such down time, I'm going to share my experience while migrating a 25Tb database with heavy load and a requirement of not more than 5minutes of down time of the service. I'm going to talk about the tools we used, problems found in the road and opportunities taken.