We consider some natural strengthenings of the well-known notion of superstrong cardinal, looking at their corresponding C⁽ⁿ⁾-versions as well, studying their properties and their connections with other usual large cardinals. In particular, we introduce the notions of C⁽ⁿ⁾-ultrastrongness and of C⁽ⁿ⁾-global superstrongness. As it turns out, the former is closely related to C⁽ⁿ⁾-extendibility, a rather robust large cardinal assumption that has found applications in other mathematical areas, while for the latter, among other things, we show that appropriate Laver functions exist, making it the second known example of a C⁽ⁿ⁾-hierarchy that has this feature.

The authors are grateful to Joan Bagaria for many helpful conversations. Further gratitude goes to the anonymous referee for his/her valuable comments and suggestions.