For each
Consider the function $f:n\mapsto k$, where the power set $2^{\aleph_n}$ has the form $\aleph_{\omega\beta+k}$ for some natural number $n$, consider k$.
This function is everywhere defined, since the power set $2^{\aleph_n}$, which 2^{\aleph_n}$ must be $\aleph_\alpha$ for some ordinal $\alpha$, and every such ordinal can be expressed uniquely expressed in the form $\omega\beta+k$, where \omega\beta+k$. The number $k$ is a natural number, simply the residue of $\alpha$ modulo $\omega$, the finite part of $\alpha$ sticking above its last limit.
Consider the corresponding function $f:n\mapsto k$, where $2^{\aleph_n}=\aleph_{\omega\beta+k}$ as above.
This So this function is everywhere defined at each $n$.
But meanwhile, but we cannot provably determine in ZFC say with certainty any particular value of it$f$. If the GCH holds, then $f(n)=n+1$, but if the GCH fails in complicated patterns, then $f$ will be similarly complicated. And neither can we do so
We cannot provably determine in ZFC---or even in much stronger theories , such as ZFC plus + large cardinals. cardinals---any particular value of $f$. Indeed, the particular values of $f(n)$ are completely independent from one another, from the perspective of what is provable in ZFC or in ZFC+large cardinals, since by Easton's theorem it is consistent with ZFC that $f$ is any function at all from $\mathbb{N}\to\mathbb{N}$. For any particular function $g:\mathbb{N}\to\mathbb{N}$, there is a forcing extension of the universe whose version of $f$ coincides with $g$.
So this would seem to be a fairly strong sense in which we do not know the values of $f(n)$ for any particular $n$.
