Powering threshold for an endomorphism of a group

Definition
Suppose $$G$$ is a group and $$\sigma$$ is an endomorphism of $$G$$. The powering threshold for $$\sigma$$ is defined as the powering threshold of the sequence of groups $$G_i = \sigma^i(G)$$, i.e., the sequence:

$$\sigma(G),\sigma^2(G),\sigma^3(G),\dots$$

In other words, it is the largest positive integer $$m$$ such that $$\sigma^i(G)$$ is powered for all primes less than or equal to $$i$$.

Note that if the condition that "$$\sigma^i(G)$$ is powered for all primes less than or equal to $$i$$" holds for all $$i$$, the powering threshold is $$\infty$$, and such an endomorphism is termed an infinitely powered endomorphism.

Definition for rings
The definition above can be applied to any additive endomorphism of a non-associative ring.