Determining the response of a medium whilst in the presence of a tracer particle (TP) driven by an external force is a ubiquitous problem. It is for example at the heart of active micro-rheology, which has become a powerful experimental tool for the analysis of different systems in physics, chemistry, and biology . At the theoretical level, it has been the subject of many studies, most of which focus on analysing the velocity of the TP. Behavior beyond this force-velocity relation has been addressed in recent Molecular Dynamics simulations of active microrheology of glass-forming liquids and revealed superdiffusive fluctuations associated with the position of the TP. Such anomalous response, whose mechanism remains elusive, has been shown to occur only in systems close to their glass transition, suggesting that this could be one of its hallmarks. I will show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. I will rely on an analytical solution of a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. This model represents a combination of two paradigmatic models of non-equilibrium statistical mechanics; asymmetric (for the TP) and symmetric (for the host particles in the system) simple exclusion processes. Through examining such non glassy systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a characteristic response of crowded environments to an external force.