The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality.
Compact closed categories are *-autonomous, with the monoidal unit as the dualizing object.
A familiar example is the category of finite-dimensional vector spaces over any field k made monoidal with the usual tensor product of vector spaces.
The dualizing object is k, the one-dimensional vector space, and dualization corresponds to transposition.
The category of complete semilattices with morphisms preserving all joins but not necessarily meets is *-autonomous with dualizer the chain of two elements.
A degenerate example (all homsets of cardinality at most one) is given by any Boolean algebra (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object.
The formalism of Verdier duality gives further examples of *-autonomous categories.
For example, Boyarchenko & Drinfeld (2013) mention that the bounded derived category of constructible l-adic sheaves on an algebraic variety has this property.
Further examples include derived categories of constructible sheaves on various kinds of topological spaces.
An example of a self-dual category that is not *-autonomous is finite linear orders and continuous functions, which has * but is not autonomous: its dualizing object is the two-element chain but there is no tensor product.
The concept of *-autonomous category was introduced by Michael Barr in 1979 in a monograph with that title.
Barr defined the notion for the more general situation of V-categories, categories enriched in a symmetric monoidal or autonomous category V. The definition above specializes Barr's definition to the case V = Set of ordinary categories, those whose homobjects form sets (of morphisms).
Barr's monograph includes an appendix by his student Po-Hsiang Chu that develops the details of a construction due to Barr showing the existence of nontrivial *-autonomous V-categories for all symmetric monoidal categories V with pullbacks, whose objects became known a decade later as Chu spaces.