Higher arity operations appear quite naturally when homotopy theory enters the stage; e.g., [$A_\infty$-algebras][1], [$L_\infty$-algebras][2] and [$E_\infty$-algebras][3].


  [1]: http://ncatlab.org/nlab/show/A-infinity-algebra
  [2]: http://ncatlab.org/nlab/show/L-infinity-algebra
  [3]: http://ncatlab.org/nlab/show/E-infinity+algebra