Question

A differential-geometric example of such a thing would be differential forms (with operations of addition, wedge product and differentiation). More generally, one considers DGLAs, Differential Graded Lie Algebras (with the usual caveat that the Lie bracket is not quite commutative/associative); the operations are addition, derivation and the bracket (as well as multiplication by scalars as a free bonus). The main example is, of course, differential forms on a manifold with values in a Lie algebra. One uses DGLA's to describe deformations of pretty much anything under the sun, look here.