Well, if all you care about is the space complexity you could do the same thing with 8 bytes of memory, an integer and a floating point. Start the floating point at zero. For each item, set the integer to 0, then compare the item to every other item (including itself) and add one to the integer if they are equal. After each iteration, add one divided by the integer to the floating point, and repeat for the next item. When you're done the answer will be stored in the floating point.

Of course this method requires O(N^2) time (that is, 1,000,000,000,000,000,000 compare operations when counting a billion objects), but who cares? It only uses 8 bytes!