exp(x)1 definately has noninteger iterated; see heirarchies of height 1/2 at http://tetration.org/Combinatorics/Schro...index.html which is listed in the OEIS as A052122. I don't have access to my computer, but it looks like our results for heirarchies of height 1/2 agree. I also have the general solution which checks with OEIS entries for heirarchies of height 2, 1, 1/2, 1, 2, 3 and 4.
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Iterability of exp(x)1

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