Are there any reasonably natural algebras whose product (bracket) almost, but does not quite, satisfy the Jacobi relation?

A priori it doesn't matter whether the bracket is anti-symmetric.

The question is deliberately vague about "almost, but does not quite", just to see if this strikes any chord. It can mean that the failure to satisfy Jacobi has a factor of epsilon, so that as epsilon goes to zero you get a Lie algebra.