First a trivial remark: if you have a binary operation you automatically have higher arity operations by nesting. Hence I would not say that there are fewer such algebras. But there is a sense in which that is cheating. Examples of these are some of the triple systems, say Lie triple systems, which are to symmetric spaces what Lie algebras are to Lie groups: namely, the best linear approximation.

Starting at least in the 1940s, the Russian algebraist AG Kurosh and his school sought to generalise many of the algebraic structures with a binary operation to an $n$-ary operation. This is explained in the paper/monograph Multioperator rings and algebras from 1969 as well as in work of Baranovic and Burgin from 1975 on Linear $\Omega$-algebras. Perhaps the best known example of this kind of structure are the $n$-Lie algebras introduced by VT Filippov in 1980.

3-Lie algebras had previously appeared in work of Nambu trying to generalise Hamiltonian mechanics by replacing the symplectic form by a closed 3-form. This line of work was continued by Takhtajan and collaborators.

In the last few years, $n$-ary Leibniz algebras (but mostly $n=3$) have been given lots of attention due to the unexpected rôle they play in the AdS$_4$/CFT$_3$ correspondence for M2-branes. Two years ago I gave some lectures on some of the underlying algebraic story at Nordita (Stockholm) and wrote them up. You may wish to peruse them for the references.