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Thank you, Daniel, for hinting to this interesting site!
I didnt know it before and think this site is a very beneficial idea (for humankind or at least the mathematically interested humankind).
My favourite is:
Does the exponential function have a square root?

I will give a comprehensive answer as soon as I got the time.

edit: Oh I see its similar to:
What’s a natural candidate for an analytic function that interpolates the tower function?
Where some people from our forum gave some hints already.
Math Overflow has added a fractional-iteration tag we can now use to make and track questions of interest to us.