Infinite finitely generated amenable periodic groups I know that the Grigorchuk group is an example of this. I also know that there are other Grigorchuk groups that satisfy this as well. Are there any other examples? Is any general structure/classification result known?
 A: Nekrashevych has recent results on this, but either they are behind a paywall (and unclear from the abstract), or they are not available yet. (This paper is clearly related, but not about torsion groups.) Anyway, he recently (May 2019) gave a talk on this stuff, and the title and abstract are below (thry can be found here). Interpreted for your question, his work says that every known example of amenable Burnside groups are related to Grigorchuk's group (he referred to these as "bottom up" rather than the "top-down" small-cancellation-based approach of, for example, Ol'shanskii when he constructed the Tarski monster groups).
Title. Amenable torsion groups
Abstract. The classical methods of constructing groups of Burnside type (infinite finitely generated torsion groups) typically produce non-amenable groups. For example, it is known that all Golod-Shafarevich groups and all known groups of bounded exponent are non-amenable. The only exceptions are some groups generated by automata (e.g., the Grigorchuk group). We will discuss a new class of groups of Burnside type constructed using topological dynamical systems and etale groupoids. A rough idea of the method is "deforming" a locally finite group to produce a finitely generated group, but preserving some of the finiteness condition of the group. This class of groups includes the known examples of amenable groups of Burnside type, but it also includes many more examples (e.g., simple groups and new groups of intermediate growth). Some open problems and directions for further research will be discussed.
