assume $ x in[0,1]$and restrict all the functions of $ x $ that we consider in this area. Consider a sequence $ mathcal S_n $ of function sets, where $ n ^ { text {th}} $ element is the set of all functions built from $ n $ instances of the variable $ x $ using exponentiation and parentheses. For example,

$$ small

color {gray} { mathcal S_1 =} left {x right } color {gray} {, quad mathcal S_2 =} left {x ^ x right } color {gray} {, quad mathcal S_3 =} left {x ^ {x ^ x}, left (x ^ x right) ^ x right } color {gray} {, quad mathcal S_4 =} left {x ^ {x ^ {x ^ x}}, x ^ { left (x ^ x right) ^ x}, left (x ^ x right) ^ {x ^ x}, left ( left (x ^ x right) ^ x right) ^ x right } color {gray} {,} \ small color {gray} { mathcal S_5 =} left {x ^ {x ^ {x ^ xx}}}, x ^ {x ^ { left (x ^ x right) ^ x}}, x ^ { left (x ^ x right) ^ {x ^ x}} , x ^ { left ( left (x ^ x right) ^ x right) ^ x}, left (x ^ x right) ^ {x ^ {x ^ x}}, left (x ^ x right) ^ { left (x ^ x right) ^ x}, left (x ^ {x ^ x} right) ^ {x ^ x}, left ( left (x ^ x right } ^ x right) ^ {x ^ x}, left ( left ( left (x ^ x right) ^ x right) ^ x right) ^ x right } color {gray} { , , text {etc.}} $$

Note that different parentheses may still result in identical functions on $[0,1]$ – we consider them as the same function (it appears only once in the set). The cardinality of the elements $ left | mathcal S_n right | $ is counted by the $ small text {OEIS} $ sequence A000081 $which has been pretty well studied.

We assume that the value of a function on $ x = $ 0 is the right limit of the corresponding expression for $ x to0 ^ + $ (the limit can be either $ 0 or $ 1 $, the numbers of each result are counted by the $ small text {OEIS} $ sequences A222379 $, A222380 $).

For each set $ mathcal S_n $ I have integrated each function $[0,1]$, sorted the list of results by order of magnitude and plotted them (the indexes in the list have been resized to fit on $[0,1]$, and discrete points connected by a polygonal chain). It seems like $ n $ increases, the graph begins to appear "smoother" and converges to a certain line. For example, for $ n = $ 8, the graph looks like this:

And for $ n = $ 11 it looks like this:

Can we (prove) that the sequence of graphs indeed converge everywhere on $[0,1]$? If yes, is the limitation line continuous? It's smooth?