I've noticed that often after a ball goes out a different ball is used. Also I don't know if the same ball will be used for both halves (plus extra time). So how many balls are used during a typical FIFA World Cup 2014 game?
FIFA's laws continue to state that
'The ball may not be changed during the match without the authority of the referee.' However as you may have seen during the FIFA World Cup as well as other competitions, many associations have accepted the Multiball system. So it is hard to find a documented maximum number or an average number of balls per match. But as per the wikipedia entry on the Multiball system, generally about a maximum of 7 balls are used per game.
Also, I read here that Adidas manufactured and tested 20 balls for the Final World Cup game between Germany and Argentina.
A lot of people are against this system because home teams, specially in England use it to their advantage and ask the local ball boys to waste time during away throw-ins, etc.
This article from The Guardian says that:
In total 3,240 balls were used throughout the tournament (2014 FIFA World Cup Brazil), 20 for each match
Generally, FIFA Laws of the Game Law 02 - The Ball: 3. Additional balls only states that:
Additional balls which meet the requirements of Law 2 (technical requirements) may be placed around the field of play and their use is under the referee’s control.
The accepted answer incorrectly references to Law 02 - The Ball: 2. Replacement of a defective ball.
The ball may not be changed during the match without the referee’s permission.
The following statement applies only to defective balls, as it is written under "Replacement of a defective ball section", and it doesn't apply to any balls.
E.g. if suddenly a player realizes that the ball he was playing with is defect, he has to notify the referee first and then change the ball, but if he changes a normal ball during throw-in, there's no need to notify the referee.