The answer is 43. (make one number 43, and all the rest 1)
Work:
The mean would be one seventh of the sum of the numbers she picked.
If we want the mean to be 7, then the numbers she chose must add up to 49.
She has to pick positive (non zero) whole numbers. So, the lowest number that she can pick is 1.
So, to have the largest possible value for one of the numbers, the six others should be 1.
The largest possible value for one of the numbers is thus 49-6 = 43.
