Some of the most fundamental puzzles in theoretical physics can be phrased as puzzles of technical naturalness.  Examples include the cosmological constant problem, the Higgs mass hierarchy problem, the linear scaling of resistivity with temperature in the strange metal phase of high-tempterature superconductors etc.  I will re-examine the notion of technical naturalness in systems whose short-distance behavior exhibits nonrelativistic symmetries of "Aristotelian spacetime", pointing out various surprises and novelties, with possible applications in particle phenomenology, cosmology and condensed matter.