I don't really know 3/5ths of f. all about the rules of golf. And as already stated by people that do know the rules, it is against them. In regards to the discussion whether playing additional balls would improve your score and handicap, I think it would.
My take on it...
Even if you're only scoring the first ball you hit, that first ball you hit may not be the first next shot you take. If it's not, then you're probably giving yourself an advantage on your "first" ball score.
Easiest example to explain what I mean would be in terms of testing the conditions in a bunker:
You tee off more than one ball
You hit two of them into the same large greenside or fairway bunker on the same hole
You don't hit the "first" ball shot first, you play the "second" ball first as it's further from the pin
BOOM, you've tested the condition of the hazard and given yourself an advantage for your "first" ball, the scoring ball.