I did a calculation which says that every application of SHA-256 reduces entropy by about 0.5734 bits. I have no idea if that's correct.
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.
I took it one step further and considered the distribution among the outputs, which is not uniform; the result for the amount of entropy lost is (1/e)*sum(log_2 n / (n-1)!). But this is probably little more than a nice exercise, as SHA-256 is likely too far from random for this calculation to be meaningful.
And I mistakenly input ln rather than log_2 to the software, so the value I want really should be 0.827.