Question is in the title. For bonus points, construct the group $G$ such that it also has no infinite proper subgroups. (This second question relates to the Prüfer group, but that group is abelian, and clearly $G$ is nonabelian since it has nonabelian subgroups.)
Ignoring the second constraint for now, it is clear that the direct product of every finite group contains every finite group as a subgroup, but it is not a very "natural" group. Are there any examples of more common infinite groups that also happen to have every finite group as a subgroup?