Puzzler's Edge

Great job, everyone!

Now we have another puzzle.

Puzzle 2:

Formalities aside, let's get started.

You know about that whole "adding" thing - you can put 2 and 2 together to make 4.
But when you add tons, you just say how many times you add. Then you can say "x times 2."
When you just have these two, it's easy to say how you can add in weird amounts - for example, adding 1.2 times, by using multiplication.

Then comes along multiplying in all sorts of amounts - exponentiation. Euler even made a way to multiply complex amounts of times but we won't really care about that so much here.

What we don't know is this -
Can you exponentiate in non-integerial amounts?

As a preliminary,
Is there a function A(x) such that for any real x, A(A(x))=exp(x)?

In bigger math words,

SpoilerShow

Can iterated exponentiation be analytically extended to the real numbers?

If not, does there exist a meromorphic function everywhere differentiable except at its poles that is equal to iterated exponentiation in integer domains?

(Iterated exponentiation defined here is base first: For example,
If we call this function "Tetration," then let us define "a tetrated to b" as:
a^(a^(a^(a^(...such that there are b amount of a in this power tower...))))

Admittingly, after a point, the power ceases to matter - the base, whether real or complex, generates a surface which determines the movement of the exponent in some domain. Through investigating this structure as a conformal map I hope to have some advancement.