Do well-known hash functions have any "impossible" output values?
No, it is not known that any cryptographic hash in common use has impossible output values (that is a bitstring of the appropriate size to be an output, but that is not reached by any input message). That's at least for MD5, the SHA(-1/-2) family of FIPS&nbsp;180-4, SHA-3 of FIPS&nbsp;202, RIPEMD-xxx, Blake/2, Tiger/2, Whirlpool.

In fact, it would be surprising that such impossible output value exists, and even more that it could be exhibited:


For a hash modeled as a random function, the probability that there is such value is lower than $1/2$ for a $n$-bit hash when there are more than $n\,2^n$ inputs; and that probability lowers with larger message input sets. See this for details.
For a Merkle-Damgård hash with Davies-Meier round function, the message block at most twice as large as the output, and the length padding at most about a quarter of the block size (like MD5 and direct larger successors are), under the assumption that the underlying block cipher is a Pseudo Random Permutation, and by an adaptation of the above reasoning, it is very probable that all outputs are reached with a single message block; and even more probable that they all are reached with two message blocks. Anything else seems to require a severe defect of the underlying block cipher, and any known such defect seems considerably lesser.
For the sponge construction of SHA-3, a similar argument can be made that a severe defect of the sponge function would be required to have some output values unreachable.