Let me point out $n$-ary operations that appear in my own research.

An algebra $(X,*)$ is said to be self-distributive if $x*(y*z)=(x*y)*(x*z)$ for all $x,y,z\in X$. The notion of self-distributivity can be generalized to identities involving operations of higher arity. We say that an $n+1$-ary operation $t:X^{n+1}\rightarrow X$ if self-distributive if $$t(x_{1},\ldots,x_{n},t(y_{1},\ldots,y_{n},y))$$ $$=t(t(x_{1},\ldots,x_{n},y_{1}),\ldots,t(x_{1},\ldots,x_{n},y_{n}),t(x_{1},\ldots,x_{n},y)$$ (see here and here).

The notion of a Laver table can be extended to $n$-ary operations. You may compute the $n$-ary Laver tables here and here.