Another way I was thinking, that still has the periodic problem, but is more natural because asymptotically it looks like tetration.

Where,

And asymptotically,

So that logarithms may behave better in the complex plane here. I wouldn't worry too much about periodicity because we can always think of the logs collecting 's. This way might actually work with holomorphy. Can't be sure though. It definitely constructs tetration on . But now, we have to deal with poles when . With this we should expect,

which looks ripe for convergence in the complex plane.

Where,

And asymptotically,

So that logarithms may behave better in the complex plane here. I wouldn't worry too much about periodicity because we can always think of the logs collecting 's. This way might actually work with holomorphy. Can't be sure though. It definitely constructs tetration on . But now, we have to deal with poles when . With this we should expect,

which looks ripe for convergence in the complex plane.