I have asked the same question previously on stackexchange without any answer (https://math.stackexchange.com/questions/923638/periodic-group-with-bounded-subgroups):

I am looking for infinite periodic groups $G$ (by periodic I mean that every element has finite order), whose finite subgroups are not arbitrarily large. So there is a constant $M$ such that for any finite subgroup $H$ of $G$ one has $|H| \leq M$.

A class of groups that satisfy these requirements are the Tarski monster groups. Here one can take $M = p$ for some prime $p$.

Are there any other notable examples or classes of examples?