Consider the infinitely nested expression
$$x=\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$$
where $\Gamma$ is the Gamma function.
Imitating the standard method for solving infinitely nested radicals, we can write
$$\Gamma(1+x)=x$$
and solve for $x$. This yields two positive solutions: $1$ and $2$.
If we instead imagine that the "$\dots$" part of the "innermost term" (hand waving here) disappears then that term becomes $\Gamma(1)=1$, the surrounding term becomes $\Gamma(2)=1$ and the hierarchy collapses until $\Gamma(2)=1$ remains.
What, then, about the "solution" $x=2$ we derived using the first method? Is either of these solutions valid? Or is it impossible to assign a unique meaningful value to the infinite expression in the first place?