# Integrals of power towers

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?