We present the two-photon absorption (2PA) spectra of a series of conjugated porphyrin dimers and show that they possess extremely large intrinsic (femtosecond) peak 2PA cross sections, up to sigma2 = 1 x 104 GM in the near-IR region; these are among the highest values measured for any organic molecule. Moreover, we demonstrate that the second-order perturbation theory applied to a simple three-level model gives a perfect quantitative description of the observed 2PA cross section. By comparing all the factors of the three-level model for dimers with those of corresponding monomer (for which sigma2 = 20 GM), we explain an approximately 500-fold cooperative enhancement in sigma2 and find that the most important factor is the strength of excited-state transition. The matrix element of dipole moment of this transition amounts gigantic values of 30-40 D for conjugated porphyrin dimers, which can be accounted for a large delocalization radius (large electron-hole separation) in this state. We also demonstrate efficient generation of singlet oxygen upon one- and two-photon excitation of these porphyrin dimers, which can be useful for two-photon initiated photodynamic therapy of cancer.