Amalgams of finite groups provide another example. Let $A$ and $B$ be finite groups and let $C = A \cap B.$ Suppose that $P$ is a Sylow $p$-subgroup of $A$, and that $C$ contains a Sylow $p$-subgroup of $B.$ Then the amalgam $A*_{C}B$ has a unique conjugacy class of maximal finite $p$-subgroups, but is an infinite group as long as $C$ is proper in both $A$ and $B$. In fact, the process an then  iterated to the case where $A$ and $B$ may themselves be amalgams of finite groups of this type, and so on. For general results on amalgams, see J.-P. Serre's book "[Trees](https://doi.org/10.1007/978-3-642-61856-7)". For applications of this type of construction to fusion systems on finite $p$-groups, see two recent papers of mine in Journal of Algebra ([Amalgams, blocks, weights, fusion systems and finite simple groups](https://doi.org/10.1016/j.jalgebra.2007.05.010)) and Transactions of the AMS ([Reduction mod $q$ of fusion system amalgams](https://doi.org/10.1090/S0002-9947-2010-05182-7)).