No, because the assumptions I made become less true the more rounds are done (maybe they're not even accurate enough after one round). The set of all possible images of SHA256^N becomes smaller for larger N until it converges to a fixed set (which is probably very large). Then SHA-256 is a permutation (one-to-one mapping) on this set. (This is true for every function from a space to itself).
Ah right, yeah. It would be interesting to know (or at least have a reasonable estimate) how large this conversion set is.
Intuitively I'd say it's quite sufficient to hold as a secure hash for the forseeable future, although I don't have any reasonable arguments for this.