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][1] and [here][2]).

The notion of a Laver table can be extended to $n$-ary operations. You may compute the $n$-ary Laver tables [here][3] and [here][4].


  [1]: https://arxiv.org/pdf/1403.7099.pdf
  [2]: http://www.ieja.net/files/papers/volume-17/4-V17-2015.pdf
  [3]: http://boolesrings.org/jvanname/lavertables-computation-endomorphic-ternarylavertablecalculator/
  [4]: http://boolesrings.org/jvanname/lavertables-computation-endomorphic-fulloutputternarylavertablecalculator/