I know of Tarski monsters and the Burnside Problem. I would like to know if there is an infinite finitely generated group $G$ such that for any $g$ and $h$ in $G$, the subgroup generated by $\{g,h\}$ is finite.

I am also interested in related questions:

For which $(m,n)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which is finite?

For which $(m,n,k)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which has cardinality at most $k$?

Thank you.

Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012) $\endgroup$ – Moishe Kohan Feb 1 '17 at 1:12