[morelisp]

The mathematical term for this is a "fixed point", where f(x) == x.

Assuming a perfectly random uniform distribution, the usual desirable property of a cryptographic hash - the probability of a hash function not having a fixed point (that is, hashing at least one x to itself) is (1-1/n)**n, where n is the number of possible outputs. As n approaches infinity - which it does pretty rapidly in this case, since we're talking about 2**32 to 2**512 in practice - this approaches 1/e, or about 37%.

So, not only is it possible, but most "good" hash functions (63% of them) will have them.