some general thoughts

[Gottfried]

I think, it has even the characteristic of a paradigm-change to move from the view as "appending exponents" to the view of "appending bases" - this seems to be much more important than the difficulties of introducing another notation. The question is: how do we embed our view and handling of tetration in the case of infinite height in the very basic and important concept of approximation via partial evaluation (partial sums).

"paradigm-change" is perhaps a bit exaggarated.
I didnt know of anybody who thinks of tetration as appending exponents. The right-bracketing rather forces one to use appending of bases, which is equivalent to \( {^nb}=f^{\circ n}(1) \) for \( f(x)=b^x \).
I also dont know what you mean by different partial evaluation, we already know that \( {^nb}=\{b,1\}^n \) in your notation. So (finite) tetration is just a particular case of appending bases. So the partial evaluations of \( {^\infty x} \) is exactly about appending bases. Or what do you mean by partial evaluation?

In a sci.math thread I posed this question, when I started metabolizing the fixpoint idea: what is sqrt(2)^sqrt(2)^... =2? = 4?
(I can't find the google-id, here is another one)
And then showed, that the initial assumption of any of these possibility could be valid, and that this made me think....
The argument was, that partial evaluation had to proceed *always* from sqrt(2), (sqrt(2)^sqrt(2)), ... and this way converges to 2 and thus no other possibility occured. (Either by argument or by my own mind only the analogy to partial sums of inifnite series appeared) That was made as a very important point, otherwise we would introduce inconsistencies.
I finally agreed to the arguments, but the discussion put a seed in my thoughts which evolved then to the view, that it is really a different *definition* or -to make it more explicite- a different paradigm in regard to the view of tetration. (well - may be this expression is a bit harsh)