In dimension $1$ the most common reason for the non-existence of a biholomorphic mapping of some unbounded domain to any bounded one is the Liouville theorem (any bounded mapping turns out to be constant), however, a classical construction by Ahlfors and Beurling shows that there are non-Liouville unbounded planar domains which still fail to be biholomorphic to a bounded one (such domains allow nonconstant bounded holomorphic function but no injective bounded holomorphic ones). My question is about the higher dimensional situation. Do there exist unbounded domains which are not biholomorphic to any bounded domain for reasons other than the Liouville theorem? To exclude trivial cases, such as the Cartesian product of a Ahlfors-Beurling domain with $\mathbb C$, let me state my question otherwise: If a unbounded domain allows a mapping to a bounded domain which is onto and no point in the image has, say uncountable preimage (or maybe contains a Liouville set) does it follow that the domain is biholomorphic to a bounded domain? If the bounded holomorphic functions do not separate points but locally separate them then the answer would be no, but is this possible in the domain case? The only instances I know when bounded holomorphic functions fail to separate points are exactly the cases when one can employ the Liouville theorem in some form.