A Jónsson algebra is an infinite algebra J in a countable algebraic language that has no proper subalgebras of the same cardinality as J. We study Jónsson algebras in one particular variety in depth: the variety of Jónsson-Tarski algebras. We prove the existence, possible cardinalities, and number of Jónsson algebras in this variety. By doing so, we gain insight into Jónsson algebras in general, the varieties they can inhabit, and the cardinalities they can have.