Assuming SHA-256 is a random function (maps every input to a uniformly random independent output), you will end up having (on average) (1-1/e)*2^256 different outputs. This indeed means a loss of entropy of about half a bit. Further iterations map a smaller space to an output space of 2^256, and the loss of entropy of each further application drops very quickly. It's certainly not the case that you lose any significant amount by doing 1000 iterations or so.