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 ɛ =(T -Tc) /Tc . Upon decreasing ɛ , Gaussian approximation is violated due to local fluctuations of pairing fields that become relevant at ɛ ≤ɛ1≪1 . Characteristic scale ɛ1 is much larger than the width ɛ2 of the thermodynamical critical region, that is determined via the Ginzburg criterion, ɛ2≈ɛ1d . We argue that in the intermediate region ɛ2≤ɛ ≤ɛ1 , 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 ɛ ≤ɛ2 ; in particular, conductivity occurs to be strongly inhomogeneous in real space.