  Let $k$ be an algebraically closed field of characteristic zero, and let
$\mathcal{C} = \mathcal{R}-mod$ be the category of finite-dimensional modules
over a fixed Hopf algebra over $k$. One may form the wreath product categories
$\mathcal{W}_{n}(\mathcal{C}) = (\mathcal{R} \wr S_n)-mod$ whose Grothendieck
groups inherit the structure of a ring. Fixing distinguished generating sets
(called basic hooks) of the Grothendieck rings, the classification of the
simple objects in $\mathcal{W}_{n}(\mathcal{C})$ allows one to demonstrate
stability of structure constants in the Grothendieck rings (appropriately
understood), and hence define a limiting Grothendieck ring. This ring is the
Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$. We
give a presentation of the ring and an expression for the distinguished basis
arising from simple objects in the wreath product categories as polynomials in
basic hooks. We discuss some applications when $\mathcal{R}$ is the group
algebra of a finite group, and some results about stable Kronecker
coefficients. Finally, we explain how to generalise to the setting where
$\mathcal{C}$ is a tensor category.
