Let us define these two sequences as follows:
$a_0=1$, $b_0=c$
$a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$
$b_{n+1}\ne b_n$ for any $n$.
$x,c\in\mathbb C$
Is it possible for $a_n$ to diverge but $b_n$ to converge under these conditions? For example, if $x=2$ and $c=i$, we have
$b_1=2^i=\operatorname{cis}(\ln2)$
$b_2=2^{\operatorname{cis}(\ln2)}=2^{\cos(\ln2)}\operatorname{cis}(\ln2\sin(\ln2))$
etc.
I haven't much clue as to whether it is the case that $b_n$ can converge when $a_n$ diverges, and I can hardly work out if $b_n$ converges with $x=2$ and $c=i$.