This question arised after I recently stumbled upon the paper ["Triple groups and Cherednik algebras"](http://arxiv.org/abs/math/0304186). Doubly affine Hecke algebras are sort of a natural object to consider after finite and affine Hecke algebras. This makes one wonder, why are there no "triple affine Hecke algebras"? Or, if such a construction exists, why are they not useful? (The theory of doubly affine Hecke algebras has proved to have deep consequences and relations with many fields of mathematics, see this [previous question](https://mathoverflow.net/questions/6517/double-affine-hecke-algebras-and-mainstream-mathematics).)