  This paper describes a systematic method of numerically computing and
indexing fixed points of $z^{z^w}$ for fixed $z$ or equivalently, the roots of
$T_2(w;z)=w-z^{z^w}$. The roots are computed using a modified version of
fixed-point iteration and indexed by integer triplets $\{n,m,p\}$ which
associate a root to a unique branch of $T_2$. This naming convention is
proposed sufficient to enumerate all roots of the function with $(n,m)$
enumerated by $\mathbb{Z}^2$. However, branches near the origin can have
multiple roots. These cases are identified by the third parameter $p$. This
work was done with rational or symbolic values of $z$ enabling arbitrary
precision arithmetic. A selection of roots up to order $\{10^{12},10^{12},p\}$
with $|z|\leq 10^{12}$ was used as test cases. Results were accurate to the
precision used in the computations, generally between $30$ and $100$ digits.
Mathematica ver. $12$ was used to implement the algorithms.
