  A compatible $L_\infty$-algebra is a graded vector space together with two
compatible $L_\infty$-algebra structures on it. Given a graded vector space, we
construct a graded Lie algebra whose Maurer-Cartan elements are precisely
compatible $L_\infty$-algebra structures on it. We provide examples of
compatible $L_\infty$-algebras arising from Nijenhuis operators, compatible
$V$-datas and compatible Courant algebroids. We define the cohomology of a
compatible $L_\infty$-algebra and as an application, we study formal
deformations. Next, we classify `strict' and `skeletal' compatible
$L_\infty$-algebras in terms of crossed modules and cohomology of compatible
Lie algebras. Finally, we introduce compatible Lie $2$-algebras and find their
relationship with compatible $L_\infty$-algebras.
