We calculate Aslamazov-Larkin (AL) paraconductity σAL(T) for a model of strongly disordered superconductors (dimensions d=2,3) with a large pseudogap whose magnitude strongly exceeds transition temperature Tc. We show that, within Gaussian approximation over Cooper-pair fluctuations, paraconductivity is just twice larger that the classical AL result at the same ϵ=(T-Tc)/Tc. Upon decreasing ϵ, Gaussian approximation is violated due to local fluctuations of pairing fields that become relevant at ϵ≤ϵ1. Characteristic scale ϵ1 is much larger than the width ϵ2 of the thermodynamical critical region, that is determined via the Ginzburg criterion, ϵ2≈ϵ1d. We argue that in the intermediate region ϵ2≤ϵ≤ϵ1, paraconductivity follows the same AL power law, albeit with another (yet unknown) numerical prefactor. At further decrease of the temperature, all kinds of fluctuational corrections become strong at ϵ≤ϵ2; in particular, conductivity occurs to be strongly inhomogeneous in real space.