For a subset $A$ of a (not necessarily abelian) group $G$, define the iterated product set $$A^n=\{a_1\cdots a_n \mid a_1,\ldots,a_n\in A\}.$$ Good control on the size of $A^3$ (small tripling of $A$) is enough to give control on the size of $A^n$: $$|A^3|\leq K|A|\implies |A^n|\leq K^{n-2}|A|\text{ for every }n\geq 3.$$ On the other hand, if one only knows $|A^2|\leq K|A|$ (small doubling of $A$), such strong bounds are (very) false. One example (copied from Example 2.5.2 in Tointon's Approximate Groups) is as follows: let $H$ be any finite group, let $G$ be the free product of $H$ with an infinite cyclic group generated by some element $x$. Let $A=H\cup\{x\}$. Then $$|A^2|=|H\cup xH\cup Hx\cup \{x^2\}|=3|H|+1<3|A|,$$ while $$|A^3|\geq|H\cup xH\cup Hx\cup HxH|=1+2|H|+|H|^2=|A|^2$$ is much larger than $|A|$.