The busy beaver max shifts function, $S(n)$, has known values for $n\leq4$. Is there some basic, structural reason why it's inconceivable that we will ever find $S(n)$ for $n>4$? What is so different about $n=4$ than $n=5$? Or $n=6$? Somewhere along the way there must be some fundamental difference, otherwise $S(n)$ would be, in principle, computable for all $n$, so what exactly is this difference?
The reason that no program can compute $S(n)$ is that if you knew what $S(n)$ is you could decide the halting problem - you'd know when to stop waiting. On the other hand, for each $m$ there is a program that computes $S(n)$ for all $n \leq m$ - it just uses a table.
If it were possible to prove the value of $S(n)$ for all $n$ (that is, for all $n$ we could prove $S(n) = \alpha$ for some $\alpha$) then we could compute $S(n)$ by searching through all proofs (this assumes that our proof system is valid). So for each proof system there is a minimal value of $n$ for which you cannot prove that $S(n) = \alpha$ for any $\alpha$.
Finally, the reason that we know $S(4)$ is probably because $4$ is a really small number. The number $5$ is slightly bigger, and so things get more complicated. There's no deep reason why we know $S(4)$ but not $S(5)$, just like there is no deep reason why we know the Ramsey number $R(4)$ but not $R(5)$ (though Ramsey numbers are of course computable).
Scott Aaronson discusses this here. He and his co-author find an explicit upper bound on $n$ for which $S(n)$ can be computed.
another angle, with an informal sketch of an answer, which would take a long time to technically flesh out with further research (ie it is basically a research program): there is some preliminary evidence that the limit of what is computable about the Busy Beaver function is a measure of algorithm complexity, with two refs below that hint at this direction. roughly, small TMs with very few states cannot accomplish "as much" or "as sophisticated behavior" as more complex algorithms with more states. therefore calculation of it appears also to have a deep link with Kolmogorov complexity. another way of looking at this is that what is known/computable about the Busy Beaver function also closely coincides with state-of-the-art in automated theorem proving, which (similar to technological advance) is a continually advancing frontier based on mathematical & computer science research.
 Busy beaver problem, a new millenium attack, van Heuveln et al
 On running time of the shortest problems, Batfai