"Du hast keinen großen Hunger?
1You are not very hungry? 2You are no very hungry? 3You are not really hungry?"
Whether or not to accept question three is just kind of misleading. I know both meanings of the word "really" but wasn't sure whether to click it or not and got marked down for my choice. :(
"Nicht" means not, "kein(e/en/es/er/em)" means "no" as in "not any". In German you wouldn't say this sentence with nicht, because Hunger is a noun, you put "keinen" before it. For example: I'm not tired - Ich bin nicht müde. I don't have a cat - Ich habe keine Katze (lit: I have no cat, I don't have any cats, etc.). Now, I don't know the specific rules, you may be allowed to write it with "nicht", all I know is that the standard term to use is "kein".