Extension of tetration to other branches

[mike3]

Applied to your model of branches in the limit formula I would just guess that the limit formula does not converge if you choose infinitely man branches different from the main branch.

Actually it does seem to converge. The problem is that it seems to converge to the same value for every z in such cases. I.e., converging to a constant function. There are uncountably many such limit values, yet as constant functions they are "analytically incompatible" (is that a real term?) with the function (you can't analytically continue a constant function to tetration!), so they cannot be interpreted as connected Riemann sheets or Riemann branches, and thus not as values of the tet function.

I'm not sure of a proof though that this is actually what happens, but some way or another, it has to fail.