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

[sheldonison]

Yes, that's the gist of the solution; except you've shifted \( \phi \) by one, it should be \( \phi(s+1) = e^{s+\phi(s)} \).

I concur whole heartedly there are singularities in the complex plane....

ah, yes, I have my phi shifted by 1.  With your definition the limit as s gets arbitrarily negative is phi(s)=exp(s-1), whereas the other way the limit is phi(s)=exp(s), so I can see why I didn't catch my error, since that limit also leads to a very schroeder like equation with a formal series \( F(x)=x+a_{2}x^2+a_{3}x^3+...;\;\phi(s)=F(\exp(s)) \)
as opposed to \( F(x)=\frac{x}{e}+a_{2}x^2+a_{3}x^3+...;\;\phi(s)=F(\exp(s)) \) which is a little less like a formal schroeder series ... anyway the resulting phi is the same entire function either way, just shifted by 1.