The number of Boolean functions with n arguments is 2^{2^n} which you can see as follows: Imagine the truth table of the Boolean function. It has n argument variables and therefore 2^n many rows. For each row, you may choose as result value either 0 or 1, thus for row, there are two choices, and since these are all independent of each other, you have 2^rows = 2^{2^n} many Boolean functions with n arguments.

For n=3, you have 2^{2^3} = 2^8 = 256 such Boolean functions. So, when looking at 2^{2^2} not all 2s in that formula have the same role. Better remember the general case with 2^{2^n}.