Classifying growth rates of functions via repeated iterations

This work investigates the asymptotic behavior of functions when subjected to repeated iterations. By analyzing how functions evolve under successive compositions, we classify their growth rates into distinct categories ranging from polynomial and exponential to super-exponential regimes. The study highlights how iterative dynamics can reveal structural differences in function growth that are not apparent from single evaluations. Such a classification provides a systematic framework for comparing functions based on their long-term behavior and offers insights into applications in complexity theory, dynamical systems, and numerical analysis.