nLab (n,n)-category

Following the pattern of the notion of (n,r)-category, an $(n,n)$-category is a higher category with non-trivial cells of at most dimension $n$ and none of them guaranteed to be reversible.

So this is what is usually simply called an n-category.

Note that it is possible to go on to an $(n,n+1)$-category, or $(n+1)$-poset. You can either consider than the $n$-cells are ordered, or else consider that there are irreversible $(n+1)$-cells which are indistinguishable. (Reversible indistinguishable $(n+1)$-cells are all identities and so might as well not exist.)