What's the difference between 大哥 and 大哥哥?
Referring someone as 大哥哥 (big brother) to a kid is not unlike referring a "狗" (dog) as "狗狗" (doggy) to a child. It is a kind of 'baby/ kid talk'

大哥 is a proper term for "big brother". Just like in English, depend on context, 大哥 (big brother) can mean your own elder brother, your lover, a stranger who is older than you or about  the same age as you and so on