Classifying functions via growth rates of repeated iterations

We develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations, for example, $n+2\to 2n\to 2^n\to 2^{\cdot ^{\cdot ^2}}$ ($n$ times) etc. and their inverse functions $x-2$, $x/2$, $\log x/\log 2,$ etc. Based on this idea and some regularity conditions we define classes of functions, with $x+2$, $2x$, $2^x$ in the first three classes.

We prove various properties of these classes which reveal their nature, including a ‘uniqueness’ property. We exhibit examples of functions lying between consecutive classes and indicate how this implies these gaps are very ‘large’. Indeed, we suspect the existence of a continuum of such classes.