"Natural boundary", regular tetration, and Abel matrix

[mike3]

But actually the intuitive iteration is really difficult to handle, that's why there are nearly no results about it. No convergence proof, not even a proof that \( \log_b \) is the intuitive Abel function of \( bx \) (which is a really simple function compared to \( \exp_b \) which is the interest for tetration).

I suppose the big trouble is due to the matrix inversion, right? How come it seems that all these tetration methods are so difficult to analyze?