My understanding is that Americans usually say I'm very good while the more reserved and humble1 British people usually just say I'm fine, but in essence they are the same. What I mean is that 我很好 can be translated as I'm very good, or more rarely, I'm very fine, too. But since this is so common an expression, adding the word very usually doesn't change the meaning one bit, so it is left out.
Contrary to what you said, 我好 can be used, but it is about (but not) as terse and impolite as the one word reply 好 when spoken to strangers. 我很好 is the standard form because it's a perfect three-letter reply to the three-letter greeting 你好吗？, it sounds good to the ears and also shows a kind of courtesy and respect to the one greeting you that you are willing to speak more than the one or two word alternatives I've mentioned above.
If you really want to emphasize that you feel very good, you could say 好极了！
1. Please bear with my stereotypical distinction for the sake of making an argument.