Exotic quantum phases, arising from a complex interplay of charge, spin, lattice and orbital degrees of freedom, are of immense interest to a wide research community. A well-known example of such an entangled behavior is the Jahn-Teller effect, where the lifting of orbital degeneracy proceeds through lattice distortions. Here we demonstrate that a highly-symmetrical 5d¹ double perovskite Ba₂MgReO₆, comprising a 3D array of isolated ReO₆ octahedra, represents a rare example of a dynamic Jahn-Teller system in the strong spin-orbit coupling regime. Thermodynamic and resonant inelastic x-ray scattering experiments, supported by quantum chemistry calculations, undoubtedly show that the Jahn-Teller instability leads to a ground-state doublet, resolving a long-standing puzzle in this family of compounds. The dynamic state of ReO₆ octahedra persists down to the lowest temperatures, where a multipolar order sets in, allowing for investigations of the interplay between a dynamic JT effect and strongly correlated electron behavior.