What was King George 3's last name?

British kings and queens don't have "last names" the way the rest of us do. He was a descendant of the House of Hanover, so is considered one of the Hanover monarchs, or Hanovers for short. (Similarly, the current Queen Elizabeth II is a descendant of the House of Windsor, so one might refer to the Windsors for short.)

The formal style for referring to him would have been:

George the Third, by the Grace of God, King of the United Kingdom of Great Britain and Ireland, Defender of the Faith

Princes William and Harry also do not legally have last names, but at school and in the military when other people's last names are used they have both informally opted to use the name "Wales" because their father is the Prince of Wales. Time will tell if this becomes a precedent for last names for British monarchs, but until then no monarch has a last name as we understand it conventionally.