[Math] Examples of “Monster” groups

Question

I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are:

1.) Non-amenable groups without free subgroups

2.) Groups such that $x^n=e \hspace10pt\forall x$

3.) Groups with all proper subgroups cyclic

4.) Groups such that every proper subgroup is finite and cyclic of a given order

5.) Groups such that every elements has roots of all orders

The main source I have been using so far is a survey by Mark Sapir http://arxiv.org/abs/0704.2899.

I would like additional sources. Additional properties to the ones above would also be great. Also examples that arise "naturally" (say as a group of symmetries of some nice space rather than a combinatorial construction would be great.)

Answer 1

(3) and (4) - Tarski Monsters.

EDIT - Benjamin Steinberg pointed out this works for (1) and (2) as well.