Since 3 was interesting number, we can consider 3rd self roots:
3^(1/3)^3= 3 which is kind of not interesting even if it involves 2 complex 3rd roots of unity in the process, and:
3^((1/3)^3)=3^(1/27) = 1,041528498.. Which is more interesting (and 26 complex values).
Second iterate of self root of 3 would be than:
3^((1/3)^(1/(3^(1/3)))= 1,289058025.. but also 4 complex values.
With 2:
2^((1/2)^2)= 2^(1/4) = +-1,189207115... and 2 complex values
(2^(1/2))^(1/((2)^(1/2)))= +-1,277703768...
With 4
4^((1/4)^4) = 4^(1/256) = +-1,005429901... and 254 complex values but:
(4^(1/4))^(1/((4)^(1/4)))= 1,277703768..=(2^(1/2))^(1/((2)^(1/2))) and .. complex values
Anyway, function "Second iterate of self root of x" where f(x)=x^(1/x)
f(f(x)) = x^(1/x)^(1/(x^(1/x))) has maximum at e and it is = 1,290005369...
And values for x=2 and x=4 are the same, but not only-there are infinitely many such paired points at this stage. e seems to remain the value giving max of any positive iterate of self root.
Ivars
3^(1/3)^3= 3 which is kind of not interesting even if it involves 2 complex 3rd roots of unity in the process, and:
3^((1/3)^3)=3^(1/27) = 1,041528498.. Which is more interesting (and 26 complex values).
Second iterate of self root of 3 would be than:
3^((1/3)^(1/(3^(1/3)))= 1,289058025.. but also 4 complex values.
With 2:
2^((1/2)^2)= 2^(1/4) = +-1,189207115... and 2 complex values
(2^(1/2))^(1/((2)^(1/2)))= +-1,277703768...
With 4
4^((1/4)^4) = 4^(1/256) = +-1,005429901... and 254 complex values but:
(4^(1/4))^(1/((4)^(1/4)))= 1,277703768..=(2^(1/2))^(1/((2)^(1/2))) and .. complex values
Anyway, function "Second iterate of self root of x" where f(x)=x^(1/x)
f(f(x)) = x^(1/x)^(1/(x^(1/x))) has maximum at e and it is = 1,290005369...
And values for x=2 and x=4 are the same, but not only-there are infinitely many such paired points at this stage. e seems to remain the value giving max of any positive iterate of self root.
Ivars