Infinitely powered endomorphism of a group

Definition
Suppose $$G$$ is a group and $$f$$ is an endomorphism of $$G$$. We say that $$f$$ is infinitely powered if its powering threshold is $$\infty$$. Explicitly, this means that for all natural numbers $$n$$, the image $$f^n(G)$$ is powered for all primes less than or equal to $$n$$.