The only examples I have encountered of infinite $p$-groups with trivial center employ non-elementary methods in their construction. For instance, Example 9.2.5 of Scott's Group Theory is a perfectly satisfactory example, but it requires the wreath product (which, though an invaluable group-theoretic tool, is not what I consider an "elementary method").
Does anyone know of an example (of an infinite $p$-group with trivial center) that can be constructed and proven to have the claimed properties in a way that is friendly to, say, students of a first course in group theory? Perhaps a large product of finite groups or an easy-to-describe matrix group?
(I also welcome arguments for the nonexistence of such an example!)