For the first time G2 or G2(MP2) calculations or both have been performed to calculate the acidity and deprotonation enthalpy of classical strong mineral acids HClO4, CF3SO3H, FSO3H, H2SO4, HBF4, HPO3, and HNO3. Also, the intrinsic acidities and gas-phase deprotonation enthalpies for 39 neutral strong or superstrong Bronsted acids, Bronsted-Lewis conjugate acids, and some compounds modeling the acidic clusters of zeolites were calculated using the DFT B3LYP 6-311+G** approach. DFT B3LYP method at 6-31+G* basis was used for the calculation of the intrinsic Bronsted acidities of the conjugate acids of the carborane anion CB11H12- and its mono-. hexa-, and dodecafluorinated analogues. G2 and G2(MP2) theories describe the acidities of different compounds better than DFT B3LYP//6-311+G**. However, the DFT results could also be used for the estimation of the acidity of compounds which are out of reach of G2 or G2(MP2) theory. The estimated Delta G(acid) values obtained this way can be used as the substitutes for the unavailable experimental values, especially for those (rather numerous) compounds for which the experimental determination of Delta G(acid) is very difficult. In the case of practically all considered families of compounds extremely high acidities (low Delta G(acid) values) could be reached. if the compounds were started from KF as the parent acid, then the estimated Delta G(acid) as low as 249.0 kcal/mol (for F(OSO2)(4)H) could be reached by formation of Bronsted-Lewis conjugate acids by consecutive complexation with SO3 molecules. Also very low Delta G(acid) value (Delta G(acid)(HSbF6) = 255.5) could be reached by complexation of HF with SbF5. At least as high intrinsic acidities as in case of the strongest Bronsted-Lewis superacids could be reached in the case of progressive introduction of highly electronegative, correctly oriented polarizable dipolar electron-accepting substituents into the acidity site. Indeed, the introduction of five CN groups into cyclopentadiene is expected to lead to the acidity Delta G(acid) = 250.1 kcal/mol which is lower than the corresponding quantity even for hexafluoroantimonic acid (Delta G(acid) = 255.5 kcal/mol). However, by far the strongest intrinsic Bronsted acidity (Delta G(acid) = 209 kcal/mol) for dodecafluorosubstituted carborane acid CB11F12H is predicted to exceed the intrinsic acidity of sulfuric acid by about 90 kcal/mol or by almost 70 powers of ten, whereas semi-empirical PM3 calculations suggest that the conjugate acid of the dodecatrifluoromethylmonocarborane anion CB11(CF3)(12) could be the first neutral Bronsted superacid whose acidity (deprotonation energy) is expected to be below the landmark 200 kcal/mol level. An approximate linear relationship is found to hold between the calculated gas-phase acidities of strong and superstrong Bronsted acids and the corresponding Hammett acidity functions of the corresponding neat acids. The simultaneous existence of the widely overlapping areas on the gas-phase acidity scale of neutral and cationic Bronstrd acids evidences strongly for the feasibility of the spontaneous proton-transfer equilibria between neutral Bronsted acids and bases.