What happens when the strength of interactions between atoms is turned up to infinity? What about when one mixes two types of superfluids?  In this talk, we will explore how ultracold atomic gases are used to create and probe these types of systems.  I will discuss our recent success producing and studying extreme quantum states in ultracold lithium gases including a "unitary Bose gas," in which the interactions between atoms have infinite strength, and a mixture of mutually interacting Bose and Fermi superfluids.  Finally, I will briefly propose how adding individual atomic ions to these systems could provide a microscopic probe of these quantum gases.