I've read that when asked the simple question 你好吗? ("How are you?"), answering 我很好 is the standard form, while 我好 is simply not used.
很 means "very", but 我很好 is simply translated as "I'm fine." and not "I'm very fine".
So if I want to say "I feel very good/I'm very fine", what should I say? Should I duplicate 很 or use another expression?
There is "我非常好" I'm extremely well, or I'm extremely good.
There is also "我非常开心" I'm extremely happy.
Maybe if you want to aim for something a little more subtle you could try "我很开心" I'm very happy.
If you are mentioning that you are this "good" you will likely be asked for an explanation!
Other ways of saying "I'm good" without emphasizing ("very") is "还行", "还好", "还可以", "还不错".
Generally, you can only find the "你好吗？我很好" type of dialogs in textbooks. Chinese people greeting each other generally do not say "我很好" at all.
很 does mean very but not always and not literally.
It's widely used in chinese for emphasis, for example consider 很多 which means many and not too many or in excess.
You could use 非常好 - 非常 is simply a modifier to take an adjective to the extreme, and 非常好 is a fairly common phrase.
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.
No one has mentioned this, so I'm doubting myself a bit, but 很 is likely used because 好 is a single character adjective. As I understand it, grammatically that just doesn't work very well and so 很 works as a generic first syllable. If an adjective is already two syllables (高兴) you can more safely assume that 很高兴 means "very happy".
My books haven't explained precisely why a single character adjective is bad, but I've always assumed that it is issues of ambiguity.
When you are asked "你好吗？"， that is "how are you?" or "how are you doing" in Chinese.
If you reply with "我很好"， that is directly translated to "I am very fine". though "I am dong great" is a better translation.
If you want to reply with something that is equivalent with "I am fine" in Chinese, you can simply say a single word "好" （fine） or "还好" （not bad), here “我” is omitted.
