Here (see Section 3) triangles in Euclidean plane are used to motivate the notion of DM stack (an equilateral triangle has more symmetries than a generic triangle).

Can something similar be done to motivate derived stacks? Some sort of moduli problem in Euclidean geometry (e.g. families circles, polygons, polyhedra) where the deformation space behaves stupidly on $\pi_0$-level but is nice when you go full derived (and you also need to have jumping automorphism groups, since we are talking about stacks)? The derived locus should have non-empty intersection with the stacky locus (assuming this sentence even makes sense).

Don't ask me why, I just love Euclidean geometry.

Introduction to Algebraic Stacks, which appeared in LMS Lecture Note Series Vol. 411. $\endgroup$ – RP_ May 16 at 15:58