These categories have loop spaces as nerves! The structure of a k-fold monoidal category can be seen as a weaker structure than a symmetric or even braided monoidal category. It is still sufficient to permit a good definition of operads in a k-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. The category of n-fold operads in a k-fold monoidal category is itself a (k-n)-fold monoidal, strict 2-category, and n-fold operads are automatically (n-1)-fold operads. I'll introduce a family of simple (pictorial!) examples of k-fold monoidal categories and classify some operads (of Young diagrams!) in example categories.