SOLVED on the site; the answer is yes. Toppila's 1976 Theorems 2 and 3 give a meromorphic function and an entire function with $\limsup_{r\to\infty}n(r,a)/n(r,b)=\infty$ for every pair of distinct values, finite values in the entire case (Toppila's claim page (1976)), and Gol'dberg's 1978 theorem gives an entire function with that upper limit infinite and the lower limit zero for every pair of distinct complex values (Gol'dberg's claim page (Goldberg, 1978)). The derived standing is solved, proved rather than answered: the question asks whether such a function exists, and Toppila's accepted full claim proves that one does.