We discuss existence issues for a Dirichlet problem in bounded domains with polynomial nonlinearities of critical growth in presence of an Hardy potential. Linear perturbations can produce positive solutions and we aim to construct sign-changing solutions, shaped as a tower of "bubbles" centered at 0 with alternating signs. The construction is optimal in the radial case as a very fine asymptotic analysis shows for radial towers of "bubbles", recovering and improving what is known in the non-singular case.