crossposted from MSE as suggested by Igor Sikora
Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable homotopy stuffs, cohomology, physics, and so on. As for $2$-categories, one doesn't even have to motivate them since they're all over the place.
However, I'm having a hard time motivating myself to study $(\infty,2)$-categories. I've been learning a bunch of facts about them: how the Duskin nerve can be regarded as an embedding from bicategories to the complicial sets model, how the Lack-Paoli nerve can be regarded as an embedding to a "simplicially enriched model", but in the end I can't see why we would want to deal with $(\infty,2)$-categories in the first place.
All I've seen so far is their use in low dimensional TQFT, and also as a way to encode the $(\infty,2)$-category of $(\infty,1)$-categories, both in the context of specific models as well as in $\infty$-cosmological contexts.
So (do we care, and if so) why do we care about $(\infty,2)$-categories?