I thought I'd compile some of the things I know...

[sheldonison]

As to the branch cuts; my solution probably (99% sure) has branch cuts other than the one at \( (-\infty,-2] \). This is a bit harder to identify, but I'll explain...

But! \( e \uparrow \uparrow s \) IS NOT PERIODIC. Therefore this function must be injective. But it can't be injective. As per (I forget the name of the authors, but it can be found in Milnor's Complex Dynamics) the orbits of the exponential are dense.  This means the sequence \( e \uparrow \uparrow \mathcal{N} + n \) for any neighborhood \( \mathcal{N}\subset \mathbb{C} \) is dense. This can't happen if \( e\uparrow \uparrow s \) is injective. CONTRADICTION.

Therefore \( e \uparrow \uparrow s \) CANNOT be holomorphic on \( \Re(s) > 0 \). It has branch-cuts. Or rather, the nicer way of saying this is, \( e \uparrow \uparrow s \) has zeroes for \( \Re(s) > 0 \).
...

I hope the reader can see this is not much different than what happens in Kneser's case. This is largely why I think this solution is Kneser's solution.

I'm going to call James' solution \( \text{Tet}_\phi(z) \), which is generated from James' entire function \( \phi(z) \).  And Kneser's solution \( \text{Tet}_k \), and conjecture that therefore they are not equal, although both solutions have \( \text{Tet}(z+1)=\exp(\text{Tet}(z) \) at the real axis.

Kneser doesn't have any singularities in the upper half of the complex plane.  Kneser's \( \text{Tet}_k \) is analytic except for singularities at integers \( \forall \mathbb{W}\leq-2 \).  

My understanding is that James' Tetration sould be defined in terms of \( \phi \) as follows:
\( \text{Tet}_\phi(z+k)=\lim_{n\to\infty}\ln^{[\circ n]}(\phi(z+n)) \), where k is chosen so Tet(0)=1

We can write an equation for when there are singularities for \( \ln(\ln(\phi(z))) \), by starting with 
\( \phi(z+1)=\exp(\phi(z))\cdot\exp(z+1)=\exp(\phi(z)+z+1) \)
So then taking the logarithm once
\( \ln(\phi(z+1))=\phi(z)+z+1 \)
And taking the logarithm twice there are singularities where \( \phi(z)+z+1=0 \)
There is a trivial version at the real axis, approximately at z=-1.3018, phi(z)=0.3018.  But how do we find the values for these singularities in the complex plane, other than the trivial 2pi i periodic examples?  Finding these values is complicated...  My conjecture is that there are an infinite number of complex values leading to singularities in \( \ln(\ln(\phi))) \) as \( \Re(z) \) increases where phi(z)+z+1~=0 ...  More later ...