Is there a simple characterization of ordinals that cannot be "escaped"? The ordinal $\omega$ is inescapable, in the sense that for any $N \in \omega$ it holds that if $f : N \rightarrow \omega$ is a function, then the supremum (taken in the class of ordinals) of the range of $f$ is an element of $\omega$.
Is there a simple characterization of the ordinals that, like $\omega$, are inescapable in this sense?
 A: It seems you are talking about regular cardinals (recall that every cardinal is also an ordinal; an initial ordinal).  These are the (infinite) cardinals $\kappa$ which have the property that any subset $A \subseteq \kappa$ of cardinality $< \kappa$ is bounded in $\kappa$.  (Actually, they are usually defined in terms of cofinality, which would result in an almost tautological assertion.)
Note that if $\alpha \in \kappa$ and $f : \alpha \to \kappa$, then the range of $f$ has cardinality $< \kappa$, and so is bounded in $\kappa$.
In general, $\omega$ and all successor cardinals are regular.  An uncountable regular limit cardinal is called a weakly inaccessible cardinal, and their existence cannot be proven from ZFC.

Added: One can show that only cardinals can have your property of inescapability.  If $\alpha$ is not a cardinal, then its cardinality $| \alpha |$ is strictly less than $\alpha$, and there is a bijection $f : | \alpha | \to \alpha$.  Clearly the range of $f$ is not bounded in $\alpha$.
