Abstract. Several recent proposals to embed inflation into high-energy physics rely on inflationary dynamics characterized by a strongly non-geodesic motion. This in turn relaxes the conditions of slow-roll to allow for potentials that are steep in Planck units, a welcome feature in view of the eta problem and the recently much discussed swampland conjectures. In this talk I will present a general framework to study non-Gaussianities in presence of non-geodesic trajectories in field space. This consists in deriving a non-standard single-field effective field theory with imaginary speed of sound supported (via a matching) by first-principle numerical computations. I will then highlight how non-geodesic motion in fields space naturally leads to 'hyper-non-Gaussianities', i.e. the bispectrum and all higher-order correlation functions are hierarchically enhanced for particular flattened configurations. The result provides powerful model-independent constraints on non-standard inflationary attractors motivated by the search for ultraviolet completions of inflation.