Actually, as far as I can tell from the explanation given by the gentleman who built the thing, this system does not have anything that would qualify as RAM, merely registers. Also, for the record, this device (while impressive) does not really qualify as a "CPU". It has a series of registers (which store data, either catching the result of an operation or holding information being operated upon) and an ALU (arithmetic logic unit. Basically a mass of resistors that perform simple arithmetic operations on two input values) but is lacking an FSM (finite state machine - a device that determines what operation is being performed by the hardware). What's more, it more or less lacks an operations register (the register that sends data to the FSM so that the FSM knows what state to choose). Both of these tasks are present but they are performed by the user.
Now, I'm quite sure such a thing could be implemented (if you can build a register you can build an operations register. And if you can build a register you can build RAM and store an extensive list of operations. With the things displayed here a universal turning machine could indeed be constructed), certain things, like a robust FSM or a seemingly large (say a 100 address word addressable RAM space) would require an enormous amount of physical space within the game and would easily dwarf those things already present in terms of complexity.
So, BRAVO to the man who created this but there is still a lot of room to grow before we have something that truly resembles what we call a computer.