Wikipedia answered the question for you. The sum of any two independent identically distributed gamma random variables is gamma. So if you sum two independent identically distributed gammas you get a gamma. Is that what you meant or did you want to sum two distributions that are identical but not gamma? If that is the case I beleive the answer is no. Also since you mentioned N possibly greater than 2, it is also true as others have pointed out that any two gammas that are independent but not necessarily identically distributed will sum to a gamma if they have the same scale parameters. So you can get a gamma as the sum of n iid gammas. You see if you can get 2 by summing two iid gammas you can get 4 by adding another 2 iid gammas with the same distribution as the first two and so on. The chi-square is a special case where this works. Distributions like the normal and the chi square that can be represented as the sum of n iid random variables of the same form (normal or chi-square respectively) are called infinitely divisible.