Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is not surjective, that is, $M_f(F) > 0$. Is the number $M_f(F)$ big?

If $F = {\mathbb F}_q$ is a finite field of $q$ elements and $f\in \mathbb{F}_q[x]$ is a polynomial of degree $d>0$, then a simple result shows that if $M_f(F_q) > 0$, then $$M_f(F_q) \geq \frac{q-1}{d}. $$ As $q$ goes to infinity, this lower bound goes to infinity. This suggests the following question for which I do not know the answer.

Question. Assume that $F$ is an infinite field. If $M_f(F)>0$. Is it true that $M_f(F) = \infty$? In other words, for an infinite field $F$, if $f \in F[x]$ misses one value, is it true that $f$ misses infinitely many values?

The answer is positive if $F$ is big in the sense that F is algebraically closed (${\mathbb C}$ etc), or topologically closed ($\mathbb{R}$ or $\mathbb{Q}_p$ etc) or if $F$ is a small field such as a Hilbertian field (number fields etc). Anyone has more examples or counter-examples?