1. They want to have more of something such as food or money than is necessary or fair.
(What is the positive degree of 'more'? Is it more or many? What is the part of speech of 'than'?)
2. I want more of something.
3. I want many of something.
(Are both grammatical?0
You use "many" if the noun that word is modifying is plural and something that can be counted, such as marbles, cats, houses, computers, children, and on and on.
http://grammar.ccc.commnet.edu/grammar/noncount.htm
You use "more" when you are comparing between TWO items or quantities: She has more blue clothes than white clothes. He has more books than I have.
"Than" is a subordinating conjunction:
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#than
Scroll up a little bit to read an explanation between "than" and "then" (which usually shows up as a typo, but not always!).