All notions of higher category have -morphisms, but the shapes may depend on the model (or theory) employed.
For a simplicially based geometric model of higher categories, i.e., simplicial sets subject to some filler conditions, the -morphisms are literally -cells in the sense of a simplicial set. This applies for example to quasi-categories, weak -categories in the sense of Street, and the weak complicial sets of Verity. In other geometric models, based not on simplices but on other shapes such as opetopes (Baez-Dolan), multitopes (Hermida-Makkai-Power), or -disks (Joyal), a higher category is a presheaf
Many notions of algebraic higher category, such as those due to Batanin, Leinster, Penon, and Trimble, are algebras over certain monads acting on globular sets (such as those induced by globular operads), so that each higher category has an underlying globular set . In that case, the -morphisms are the -cells of . In such globularly based definitions, every -morphism has a -morphism as its source and a -morphism as its target, and the source -morphisms and must be the same, as must the target -morphisms and .
For the purposes of negative thinking, it may be useful to recognise that every -category has a -morphism, which is the source and target of every object. (In the geometric picture, this comes as the -simplex of an augmented simplicial set.)
every (non-empty? -David R) -category
I think every. Up to equivalence, a -morphism in is given by a functor from the oriented -simplex to . As the -simplex is empty, there is a unique such functor for every ; thus every has a unique -morphism.
Also note that every -morphism has identity -morphisms, which just happen to all be the same (which can be made a result of the Eckmann–Hilton argument). Thus, the -morphism has identity -morphisms, so we don't need any object. (This confused me once.)