Just a few late thoughts

[Leo.W]

1)
approximate the difference by differentials.
such as replace difference by derivative.
or better approximations with higher derivatives and using taylor.
this could give asymptotics and boundaries that are potentially useful.

2)
 use infinite composition like the ideas from the beta method or gaussian method.

3)
use limits to get asymptotics 
However it's not always applicable, for example what if we're dealing with:
\(f(z+1)=f(z)+\frac{exp_e^{1/2}(z)}{e^z}\)

4) truncated methods. based on series expansions and/or fixpoints. not necc taylor.

5) sometimes series multisection and/or " fake function theory ".
Idk what is this by the name, I searched for this term many times before... Would u mind elaborating on it?

and combinations of those above ofcourse.

Yes, tommy, at least former 4 methods are ubiquitous in analysis.
Well what im trying to convey is my idea:
1. \(f(z)=z+g(z), g(z)=o(z)\) should guarantee \(f^t(z)=z+tg(z)+o(g(z))\), without necessity of asymptotic superfunctions
2. the generalization about 1., for example, \(f(z)=h(z)+g(z)+o(g(z))\) where \(g(z)=o(h(z))\) also guarantees an asymp series for \(f^t(z)\)
3. Any functions that can be included in the same group under (the multiplication as) composition while written in the forms of asymp series
have (asymp) series for its iterations
I'll show you more examples only generated from these ideas

Example 1
\(f(z)=z+\frac{1}{z}+\log(z)\)
We have
\(f^t(z)=z+t\log(z)+\frac{\frac{1}{2} (t (t-1)) \log (z)+t}{z}+ \frac{-\frac{1}{6} (t (t-1) (2 t-1)) \log ^2(z)+\frac{1}{3} (t (t-1) (t-5)) \log (z)+(t-1) t}{2 z^2}+ \frac{\frac{1}{2} (t-1) t \left(t^2-t\right) \log ^3(z)+t \left(t^2-6 t+5\right)-\frac{1}{2} t \left(2 t^3-11 t^2+13 t-4\right) \log ^2(z)+\frac{1}{4} t \left(t^3-22 t^2+47 t-26\right) \log (z)}{6 z^3}+\frac{t \left(t^3-18 t^2+41 t-24\right)-\frac{1}{5} t \left(6 t^4-15 t^3+10 t^2-1\right) \log ^4(z)+\frac{1}{30} t \left(108 t^4-615 t^3+910 t^2-405 t+2\right) \log ^3(z)-\frac{1}{10} t \left(22 t^4-315 t^3+820 t^2-705 t+178\right) \log ^2(z)+\frac{1}{5} t \left(t^4-65 t^3+365 t^2-535 t+234\right) \log (z)}{24 z^4}+O(z^{-5})\)

Example 2
\(f(z)=z+\sqrt{z}\log(z)+\frac{\log(\log(z))}{z}\)
We have
\(f^t(z)=z+t\sqrt{z}\log(z)+\frac{t(t-1)}{4}\log(z)^2+\frac{t(t-1)}{2}\log(z)+o(\log(z))\)

Example 3 (though well-known)
\(f(z)=z^2+c\)
Denote \(F(z+1)=f(F(z))\) F as a normal kneser-like real-to-real superfunction of f and \(F(0)=c\)
And write \(f_2(t)=2^t\prod_{n=0}^{t-1}{F(n)}\)\[f_4(t)=f_2(t)\sum_{n=0}^{t-1}{\frac{f_2(n)^2}{2F(n)f_2(n)}}\]
\[f_6(t)=f_2(t)\sum_{n=0}^{t-1}{\frac{2f_2(n)f_4(n)}{2F(n)f_2(n)}}\]
\[f_8(t)=f_2(t)\sum_{n=0}^{t-1}{\frac{f_4(n)^2+2f_2(n)f_6(n)}{2F(n)f_2(n)}}\]
\[f_{10}(t)=f_2(t)\sum_{n=0}^{t-1}{\frac{2f_4(n)f_6(n)+2f_2(n)f_8(n)}{2F(n)f_2(n)}}\]
with the pattern \(f_{2k}(t)=f_2(t)\sum_{n=0}^{t-1}{\frac{\textstyle \sum_{m=1}^{k-1}{f_{2m}(n)f_{2k-2m}(n)}}{2F(n)f_2(n)}}\)
\(f^t(z)=F(t)+f_2(t)z^2+f_4(t)z^4+f_6(t)z^6+f_8(t)z^8+f_{10}(t)z^{10}+O(z^{12})\)

Example 4
\(f(z)=ez+\log(z)+\frac{1}{z}\)
We have
\(f^t(z)=e^tz+\frac{\left(e^{t+1}-1\right) \log (z)}{e-1}+\frac{-e t+t+e^t-1}{(e-1)^2}+\frac{e^{-t} \left(e \left(e^2-1\right) \left(e^t-1\right) \left(e^{t+1}-1\right) \log (z)+(1+e)^2 \left(-e^t\right)+\left(e (1+e) (e-1)^2+1\right) e^{2 t}+e (-t+e (e (t-e+1)+2)+1)\right)}{(e-1)^3 (1+e)^2 z}+O(z^{-2})\)

Most are un-trancable, non-taylor, you may notice it's similar to our old functional iteration where f(0)=0 and f'(0)=1 and has a taylor series.
The series are precise, and not approximation, so the asymp method may be better in some cases.
This is the group structure (infinite-order though)