Something like a field but with 3 operations?

Have a look through A Course in Universal Algebra, by S. Burris & H.P. Sankappanavar. You'll find many examples of algebras with more than two operations there. It builds up a rich theory for them in fact, proving, for instance, the isomorphism theorems in full generality.

Poisson algebras come to mind. They have an associative ring structure as well as a second "multiplication" that behaves as a Lie bracket, compatible with the first multiplication.