The composition law in a monoid is usually represented using a binary operation (multiplication) and a zeroary operation (unit), but I view it more naturally as an operation (say, a bracket) taking any finite list of arguments and being associative in the sense that brackets can be eliminated in any expression, for example we have the identities
[a,[],b,[[c,d],e],[f]]=[a,b,c,d,e,f]
and
[a]=a.
Then we can define a zeroary operation 1:=[] and a binary operation a*:b=[a,b], and recover the bracket from them, using identities such as [a,b,c,d]=[a,[b,[c,d]]]. These two operations satisfy the usual axioms of a monoid, and any two operations satisfying them can be extended to an associative finitary bracket.
I view the usual representation by a binary and a zeroary operation as an artifact for being able to produce simpler-looking proofs that the structures that we encounter are monoids.
My point is that naturally binary operations are not that common either! Perhaps an example is the Lie bracket.