Attempt to find a limit point but each step needs doubling the precision...

[Gottfried]

Perhaps a silly question but assuming Gottfried and Jay plot iterations of type sexp^[k](x) for x values near the real line ...

Then why do these contours of iterations not intersect ?

I mean why do we not get contours like here :

http://math.eretrandre.org/tetrationforu...hp?tid=499

Afterall iterations of sqrt(2)^x are just as chaotic as those of exp(x) or not ?

Just to that question: no, it's different. Because b=sqrt(2) is in the interval 1 < b < exp(exp(-1)) and has two fixpoints on the real line (so all iterates of values on the real line >0 are on the real line again and we have even finite values for the iterates when h goes to infinite height (namely the two real fixpoints) while when b=exp(1) then it is outside the Euler-interval 1<b<exp(exp(-1)) and we cannot iterate infinitely on the real axis except with the result going to positive infinity as well.
For the iterations with complex heights one can consider Dmitrii's further plots in his articles on tetration with base b=exp(1)...