What gauge wire do I need to supply 15A to small building 440 ft from the main house?
Using one of the many wire size calculators on the internet, for 120v single phase: 


  1 conductors per phase utilizing a #3 Copper conductor will limit the voltage drop to 2.59% or less when supplying 15.0 amps for 440 feet on a 120 volt system.


Also, please be sure to use the actual wire length, not just the distance between buildings (eg taking into account depth in ground, height to panels, farthest plug on your circuit, actual path wire will take, etc). With the same parameters, it looks like you can go to about 510 feet before you need to use #2 wire.