Does Conway's PRIMEGAME generate all prime powers of 2?
Yes, PRIMEGAME outputs $2^k$ if and only if $k$ is prime.

Conway's original paper is well worth reading if you can track it down.  You can also find a very clear exposition in Richard Guy's paper Conway's prime producing machine (Mathematics Magazine 56(1):26–33, 1983), including the wonderful cartoon below. (Yes, that's Conway with the Alexander horns, referring to a famous drawing by Simon Fraser.)  Conway himself posted a concise proof on the math-fun mailing list.  There's also a brief explanation at the OEIS blog.