Lie algebra contractions play a somewhat interesting role in physics (e.g. giving meaning to the statement "special relativity reduces to Galilean relativity in the $\lim\limits_{c \rightarrow \infty}$" when on the face of it that doesn't make sense -- $c$ is a constant. Contractions essentially formalize how this is to be understood as a statement about the ratio of some characteristic length scales).
Out of curiosity: has any of this been generalized to Lie $2$-algebras (see e.g. here)? Or even further? Are there any references going in this direction?
Contractions (and more generally degenerations) of Lie groups and Lie algebras were generalized to many other algebraic structures, for a little survey see my article here. There is a strong relationship to deformation theory, and the deformation theory of Lie $n$-algebras has been studied, e.g. see here for references, section 11.7 and also the paper Deformations of Lie $2$-algebras . Certain (jump) deformations correspond to contractions.