Do all single variable identities over exponentiation have the same length? Consider exponentiation on the natural numbers. Suppose we have an equation involving only the variable $x$, where there are $m$ $x$'s on one side and $n$ $x$'s on another side where $m$ and $n$ are different, like $(x^x)^{x^x}=(x^x)^{(x^x)^x}$, where the brackets can go anywhere. Is it always the case that there is a natural number such that the equation does not hold? In other words, do all single variable identities over exponentiation have the same length?

Note: This only partially answers the question, as it does not consider the case $m=n$, and $m,n$ are defined slightly differently here (Although if the case $m=n$ was shown to hold too, then it would cover all cases of the original question).
To make the notation easier on the eyes, I'm going to write $(p_1,p_2,...,p_n)$ to mean $x^{p_1^{p_2^{...^{p_n}}}}=x^{\left(p_1^{\left(p_2^{\left(...^{p_n}\right)}\right)}\right)}$ where the $p_i$ are polynomials in $x$
Every equation of the form we are interested in can be written in this way, e.g. $(x^x)^{x^x}=x^{x^{x+1}}=(x,x+1)$.
Now suppose we have an equation $(p_1,...,p_n)=(q_1,...,q_m)$, where $p_i,q_j$ are not constants for all $i,j$, otherwise we can raise the term below to the integer power (we know the power will be an integer because in reducing the expressions in $x$ to this form, we are only multiplying an integer number of times).
Firstly, if $n>m$, then $p_{n-1}^{p_n}$ grows faster than $q_m$, since one is an exponential expression and the other is a polynomial expression. Then we can show $p_{n-2}^{p_{n-1}^{p_n}}$ grows faster than $q_{m-1}^{q_m}$ by taking logarithms:
$p_{n-1}^{p_n}\log{p_{n-2}}$ grows faster than $q_m\log{q_{m-1}}$ since $p_{n-1}^{p_n}$ grows faster than $q_m$.
We repeat this process to get that $(p_{n-m},p_{n-m+1},...,p_{n-1},p_n)$ grows faster than $(q_1,q_2,...,q_m)$, so certainly $(p_1,...,p_n)$ grows faster than $(q_1,...,q_m)$. So taking an $x$ big enough will break the equality.
This argument relies heavily on the fact that "exponentials beat polynomials", the idea that expressions like $a^x$ eventually grow faster than any finite degree polynomial for any positive constant $a$, so certainly it will be true if $a$ is a polynomial in $x$.
