To add to the list of examples:

1. [Heaps](http://ncatlab.org/nlab/show/heap) 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.

1. [Totally convex spaces](http://ncatlab.org/nlab/show/totally+convex+space) 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.

1. Similarly, $C^*$-algebras and there's a theory closely related to Banach algebras.  See [this page](http://ncatlab.org/nlab/show/algebraic+theories+in+functional+analysis) 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](http://ncatlab.org/nlab/show/Eckmann-Hilton+argument).  Therefore, once we start applying common identities, we find that we can often reduce the arity down to something palatable.