Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a singleton. In the meromorphic case this set of exceptional values is of cardinality at most two (including $\infty$).

**Q:** What can we say about $\mathbb{C}\setminus f({D})$ if $f\colon D\to \mathbb{C}$, $D\subset \mathbb{C}$ is nonempty open and connected, is holomorphic:

If $n\ge 3$, then does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of cardinality $n$?

Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is of infinite but countable cardinality?

Does there exist $f$ such that $\mathbb{C}\setminus f({D})$ is a meagre set or of Lebesgue measure zero but of uncountable cardinality?