  Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive
for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given
by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We
show such a sequence converges in $\widehat{\mathbb{Z}}$ and the limit is
independent of $b$, if and only if $f$ does not become a permutation of length
$p$ on $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$. We also show that
$b'$-adic asymptotic approximations of the equation $f^y(a) = y$ holds in
$\mathbb{N}$ for some bases $b'$.
