To add to the list of examples:
Heaps have a single ternary operation (identities on linked page). In short, a heap is to a group what an affine space is to a vector space: as soon as you pick an identity then you get a group.
Totally convex spaces which are spaces that allow arbitrary convex combinations. Simple examples are the unit balls of normed vector spaces, but others such as $(0,1)$ exist.
Similarly, $C^*$-algebras and there's a theory closely related to Banach algebras. See this page on the nLab where I started gathering together a few details on these.
To address the point as to why we often only use operations of arity at most 2, here's a neat little fact. Abstractly, we can consider operations of arbitrary arity with arbitrary identities, but in concrete situations the operations usually have a high level of compatibility. A common one to ask for is commutativity. This is commutativity of operations, which is ever-so-slightly different from what we normally think of as commutativity (though the two are very closely related). If we have a binary operation with a unit, then any operation that commutes with that operation (and its unit) turns out to be formed by iterating the binary operation. This is an easy generalisation of the Eckmann-Hilton argument. Therefore, once we start applying common identities, we find that we can often reduce the arity down to something palatable.