Game (or half) is over when the referee's watch indicates the allotted time and any appropriate time allowed for injury, substitution and any other suspension of the game during that half, is up. He then blows the whistle to indicate the game is over.
This means that the game is technically over a split second before he blows the whistle to indicate the end of the game (or half).
The ONLY time additional time is allowed is to take a penalty kick for an offence that occurred before the final (or half time) whistle was blown.
Contrary to popular belief, extra time is not added on to allow a "move" to come to a natural completion, nor is it added to allow the ball to cross the goal line, a corner kick to be taken, etc.