For what positive x's the number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways is not represented by the sequence A000081? Is it exactly the set of positive algebraic numbers? Is it a superset of positive algebraic numbers? Is it countable? Is $2^{\sqrt 2}$ or $\log_2 3$ in the set?



The answer to the second question is "no". Consider the unique solution $x > 0$ to the equation $x^x = 3$. By the GelfondSchneider theorem, this number is transcendental. But we have $$((x^x)^x)^x = x^{x^3} = x^{(x^{(x^x)})}$$ so that two of the parenthesizations coincide. So evidently this set contains transcendental numbers. Lots of other solutions can be similarly generated (e.g., solve $x^{(x^x)} = 4$). 

