There are two very different questions here: The maximum range of a shotgun as in how far will the shot go and the effective range of a shotgun in taking game or breaking targets.
The effective range of a shotgun is that range within which a sufficient number of pellets can be placed with adequate penetration and energy to ensure a clean kill. That's dependent upon the game being sought, the gun's gauge and choke, the size and hardness of the shot and skill of the shooter.
My general rules are these: Assuming adequate pattern and pellet size, 50 yards for birds. No doubt they can be killed further and I've done it myself once or twice. I picked that limit because I know can hit 50 yard crossing clays often enough to reassure myself that it's not luck and therefore have confidence in my ability to take a bird cleanly at that distance. I've also shot at enough clay targets in the 80-100 yard range to know that while they can sometimes be broken you are beyond a responsible range for consistent performance on live game. Shotgun pellets are not ballistically efficient and lose velocity and energy very quickly.
Buckshot isn't a favourite load for me. I've shot animals with it and patterned enough to be unimpressed with the results. With a modified choke I might use it up to 30 yards and no further. I prefer slugs. With my current slug set up, I will keep shots within 75 yards give or take 5 yards. If I went to one of the bolt action rifled shotguns with a good scope I'd extend to 125 maybe.
No doubt these are relatively modest limits and people have killed game at further distances. In hunt camps at night and on forums like this tall tales will frequently be told about ducks being shot on the other side of the horizon or geese killed with a .410 at 75 yards.
These tall tales are told in an attempt to convince others that the teller has superior shooting abilities. Having done my share of shooting and patterning these sagas usually have the opposite effect on me. Bragging about irresponsibly long shots doesn't raise anyone in my estimation. It only confirms they are someone with whom I don't want to share a blind.
As for how far shot will travel or the maximum range here's an article on the subject.
http://www.rangeinfo.org/resource_library/facility_mngmnt/design/how_far_will.htm