(x^x)^(x^x), x^(x^(x^x)), etc...

[N. J. A. Sloane]

Isn't that the subject of the following well-known article?

%D A003018 R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876.

(there are numerous references to it in the OEIS)

NJAS