Infinitely bi-torsion-free additive endomorphism of a non-associative ring

Definition
Suppose $$R$$ is a non-associative ring and $$f:R \to R$$ is an endomorphism of the additive group of $$R$$. We say that $$f$$ is infinitely bi-torsion-free if, for all natural numbers $$n$$, and all $$i,j$$ such that $$i + j = n$$, the defining ingredient::torsion-free threshold for the subring of $$R$$ generated by $$f^i(R) * f^j(R)$$ is at least equal to $$n$$.

This is equivalent to saying that the bi-torsion-free threshold for $$f$$ is $$\infty$$.