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Claim. The answer is yes, in both readings of the question. Toppila's Theorem 2, as printed on the first page of the paper: "There exists a meromorphic function $f$ such that for every pair $a, b$, $a \neq b$, we have $\limsup_{r\to\infty} n(r, a) / n(r, b) = \infty$." Theorem 3: "There exists an entire function $f$ such that for every pair of distinct finite values $a$ and $b$, $\limsup_{r\to\infty} n(r, a) / n(r, b) = \infty$." In the meromorphic case the values range over the Riemann sphere, so $a$ or $b$ may be $\infty$ and $n(r,\infty)$ counts poles. The paper counts the roots of $f=a$ in the closed disc $|z|\le r$ where the problem uses the open disc; the two counts agree at every radius carrying no root, so the upper limits over $r$ coincide (an observation here, not the paper's). The paper introduces the two theorems as answering Erdős's question, Problem 1.25 of Hayman's 1974 collection; the constructions are explicit infinite products of rational factors. The statements are as printed in the paper, whose source card is toppila_1976_counting_function_values_meromorphic_function; the proofs are not verified in this corpus.

Source. Sakari Toppila, On the counting function for the $a$-values of a meromorphic function, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 565--572, DOI 10.5186/aasfm.1976.0235. The record dates the article to the year alone, and the page is named by that date.

Acceptance. Refereed: the paper appeared in the Annales Academiae Scientiarum Fennicae. Reviewed: the site's curator, T. F. Bloom, labels the problem solved and credits Toppila and Gol'dberg with constructing entire functions with the property; Hayman and Lingham's 2018 survey of Hayman's problems (library card, Update 1.25) records both entire examples and Toppila's meromorphic one. Nothing here is independently reviewed by this project.

Other constructions. Gol'dberg's 1978 entire function, whose ratios have lower limit $0$ as well, is on Gol'dberg's page.

Depends on. Nothing on the wiki.