We present a valence calculation of the electric polarizability of the neutron, neutral pion, and neutral kaon on two dynamically generated nHYP-clover ensembles. The pion masses for these ensembles are 227(2) MeV and 306(1) MeV, which are the lowest ones used in polarizability studies. This is part of a program geared toward determining these parameters at the physical point. We carry out a high statistics calculation that allows us to: (i) perform an extrapolation of the kaon polarizability to the physical point; we find αK=0.269(43)×10-4  fm3, (ii) quantitatively compare our neutron polarizability results with predictions from χPT, and (iii) analyze the dependence on both the valence and sea quark masses. The kaon polarizability varies slowly with the light quark mass and the extrapolation can be done with high confidence.