User blog comment:GamesFan2000/There aren't enough particles in the universe to solve a four-entry array/@comment-30754445-20181129104403

There's nothing really special in 4-arrays, though. The exact same thing happens with ordinary knuth arrows, and even ordinary power towers.

Generally, it is impossible to reduce an expression by more than a single level of recursion. You can reduce 4^^^4 to 4^^4^^4^^4, but reducing it to an ordinary power tower would be impossible (as that tower would have to be 4^^4^^4 = 4^^^3 fours high!).

Looking at an even smaller example, you cannot reduce 4^^4 to a number written in ordinary decimal form. The usual reason is given as "it has more digits than atoms in the universe", but the deeper reason is that tetration (x^^y) is two levels higher than ordinary decimal notation.

This principle can even be illustrated with ordinary numbers: Ordinary powers are two levels above addition. This means that generally, x^y results in numbers that are impractical to count or to tally. Here there are numerous small exceptions, as you can count to 4^4=256. But usually it is impossible to count up to x^y (f ior example, you cannot count to a googol, which is 10^100)

Back to {4,4,4,4}:

You can reduce {4,4,4,4} to {4,{4,{4,4,3,4},3,4},3,4} (that's one level down) but going an additional level (to {a,b,2,4}) will be impossible. In fact, the number nested {2,4}'s you'll need for such a complete expansion, would be exactly {4,3,4,4}!

And this is true at the higher levels as well. If I have some huge ordinal X (doesn't really matter what X actually is) then functions on the level of X+2 will (usually) be unreducible to functions on level X.

P.S.

You seemed to have made an error in the expansion, because you shouldn't have gotten 4→4→256 at any time. For a few dozen steps you should have nothing but 1's and 2's and 3's and 4's. Eventually you should have an innermost array of {4,3,1,2}  which expands to {4,4,{4,2,1,2}}.

That {4,2,1,2} is equal to {4,4,4} (finally we've got to a 3-entry array!) which is - of-course - equal to 4→4→4. The next step would be to evaluate {4,4,4→4→4}, which can be written as 4→4→(4→4→4) which is already far larger than 4→4→256.