One reason why I think that most operations have arity at most 2 is that most algebraic structures fall into the following few narrow class of structures and the algebras in these classes of structures tend to have common characteristics. The most prominent of these characteristics is that the fundamental operations in these structures are typically associative binary operations or are easily obtained from associative binary operations.
Ordered sets and lattices-These structures include linear orders, partial orders, lattices, Boolean algebras, median algebras, Heyting algebras, locales, closure systems.
Ring-like structures-Rings, fields, modules, algebras, vector spaces, linear algebra, semirings, Banach algebras. These structures are generalizations of the addition and multiplication operations on the natural numbers.
Group-like structures-Groups, monoids, categories, heaps, racks, quandles. These operations are often constructed from the composition of functions or as automorphism groups.
I think that the main reason why algebraic structures tend to fall into these narrow categories relies on the following observations:
It is very difficult to build a completely new algebraic structure completely from scratch using a recursive construction which satisfies nice algebraic properties. The natural numbers with can be thought of as being built from scratch. The set of all strings together with the concatenation operation is another example of an algebraic structure built from scratch.
The mechanisms for producing new algebraic structures with interesting algebraic properties from old structures tend to generate new structures in the above classes. These mechanisms include taking automorphism groups, endomorphism monoids, (semi)-direct products, congruences, various notions of completion, various notions of extending algebras (like algebraic closure, Grothendieck groups, etc.), etc.
One can generate other examples of operations of higher arity, but these are rarely fundamental operations in algebraic structures. For example, discriminator terms and Mal'cev terms are usually not the fundamental operations of algebraic structures (the pattern algebras are an exception to this rule) and neither are operations such as the determinant.
I should mention that the few ternary structures that do appear in the above three categories are usually obtained by “forgetting” the unit or origin or sense of direction in the original structure and when one adds the unit or least element back in, one obtains the original structure. In these structures, the ternary operation amounts to having two variables to replace the binary operation and a third variable to denote the missing origin. One should therefore be cautious in referring to these structures as true ternary structures. For example, in affine spaces and heaps, one forgets the unit of a group or a vector space. However, when one adds the unit to the heap or affine space, one ends up with groups or vector spaces. Likewise, median algebras can be obtained from bounded distributive lattices by forgetting which direction is up. However, a median algebra can be easily turned into a nearly distributive semilattice simply by declaring a certain element to be the least element.
This phenomenon of obtaining ternary algebras by forgetting the origin of the algebraic structure also appears with some relational structures. For example, total cyclic orders are ternary relational structures which are obtained from totally ordered sets by forgetting the ordering of the ordered set but remembering the notion of orientation. The original linear ordering can be obtained from the cyclic ordering by applying a Dedekind cut to the cyclic ordering and this Dedekind cut plays the role of the origin or identity element in the previous examples.