We consider 2 * sinh(x).
We take the half-iterate of it by using taylor series expanded at 0.
Thus f(x) is a taylor series , f(f(x)) = 2*sinh(x).
Clearly f(x) is then a good asymptote for the half iterate of exp(x) for large x.
Let g(g(x)) = exp(x)
Then g(x) =
Lim k-> oo log log log log ...( k times ) (f( e^e^e^...( k times ) ^ x))
Clearly this can be generalized to tetration.
For a real 'r' to compute exp^[r] ( ^ is iteration ) replace f(x) with (2*sinh(x))^[r] in the above formula.
Now you have ( for x and r being real ) exp^[r](x) aka tetration. It turns out this function is nowhere analytic (for the variable x) but it is continuous and infinitely differentiable on the real axis (C^oo for fixed r and variable x).
By analogue for a realvalued b we can compute (b^(x))^[r] and find the same properties as long as b > exp(1/2).
All of this has been formally proven for b > exp(2) but Im working on the smaller bases too.
One of the main reasons is that for b > exp(2) , 2*sinh(2x) has only one fixpoint (=0) whereas for exp(1/2) < b < exp(2) you have 3 fixpoints : zero and two nonreal complex conjugates.
Below is a link to my profile at the tetration forum.
At the moment I only accept the tetration forum as a good source for tetration although you can find some info and links on wiki , I do however not accept the wiki page.
Thank you for your intrest.