I'm not sure if there's a specific question here, other than perhaps "is this object studided" / "is there a better way of looking at it"; if it's too vague for this forum I apologize.
First take the free group on two elements; this has exponential growth. If I quotient by the relations $a=1$ or $b=1$ it reduces to just $\mathbb{Z}$, which is boring (only polynomial growth rate), so I quotient by $a^2=1$. My image still contains a free group on two elements, the subgroup generated by $b^2$ and $ab$, so I continue with this. Now if I quotient by any of $ab$, $b^2$, $aaa$, $aab$, $aba$, $abb$, $baa$, $bab$, or $bba$, I get again a polynomial growth rate group. If I quotient by $bbb$, though, I still contain a free group on two elements, generated by $ab$ and $abb$.
Now on $\langle a,b | a^2 = b^3 = 1\rangle$ I again look for things to quotient by, in lexicographic order. $aaaa, aaab, aaba, aabb, abaa...$ all again to small-growth groups. In fact we know that the next word to quotient by can't have any $aa$ or $bbb$ factors, including when we cycle the word; so we could proceed just checking $abab, ababb, abbab...$. It seems that $ababab$ is the next word we can quotient by while preserving the exponential nature. (It now contains a free group on two elements, generated by $ababb$ and $abbabb$).
It is apparent that there is always some word we can quotient by: for any group that contains a free group on two generators $x, y$ also contains a free group on three generators $xy, xyy, xyyy$, and we can quotient by one of those. It also true that the limit of these groups is well-defined: it is just a direct limit of this series of groups.
I'm curious if anyone has any insight into the nature of this direct limit $G$. For instance, does $G$ itself have exponential growth? My intuition says no, because then I would expect to find a copy of the free group on two elements inside, and a generator of it would have been quotiented out at some finite step in the construction. I'm not confident in that statement though. If anyone might know related literature or have thoughts on this I'd be curious to hear; it seems like it could be an interesting structure.
