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.



Almost-Lie Algebras?
See Section 2.3 of the lecture notes called *Geometric Models for Noncommutative Algebras* by Ana Cannas da Silva and Alan Weinstein. There they define an "almost Lie algebra" to be something with an antisymmetric bracket but which does not necessarily satisfy Jacobi. In Section 3.2 they connect this to the notion of "almost Poisson manifold", which is just a manifold equipped with a bivector field. The bivector field defines a skew-symmetric bracket on smooth functions which may or may not satisfy Jacobi.


