$\ast$-autonomous categories with non-invertible dualizing object?

Question

1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:

A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a global dualizing object: An object $\bot$ such that the canonical morphism $$d_A: A \rightarrow (A \multimap \bot) \multimap \bot$$ which is the transpose of the evaluation map is an isomorphism for all $A \in C$.

Secondly, call an object $A$ in a (symmetric) monoidal category $(M,\otimes,\top)$ invertible if there exist an object $B \in M$, and two isomorphisms $\eta: \top\rightarrow A \otimes B$ and $\epsilon: B \otimes A \rightarrow \top$ satisfying the two zig-zag-identities. (In fact, by an argument of Saavedra Rivano it suffices to require that only one zig-zag identity holds, but that is beside the point.)

2. Question
What are ('real-world') examples of $\ast$-autonomous categories with non-invertible global dualizing object?

Answer 1

The dualizing object of a star-autonomous category is invertible iff the category is compact closed. See here.

There are lots of examples of star-autonomous categories which are not compact closed. In algebraic geometry, one systematically constructs dualizing sheaves which are not generally invertible -- some of this is alluded to in the comments.

In homotopy theory, the category of spectra of finite type is a good example. The dualizing object is the Anderson dualizing spectrum $I_{\mathbb Z}$.