In order to study the polarity effect on the critical phase behavior of fluid binary systems the critical temperatures of binary mixtures of carbon dioxide, ethane, ethene, hexane, or 1-hexene with a second component of very different polarity such as methanol, 2-hexanol, 2-octanol, 2,5-hexanediol, nitromethane, N,N-dimethylformamide, or N-methylacetamide are determined as a function of pressure up to 1000 bar. The critical p(T) curves obtained exhibit a whole pattern of different transition types between liquid-gas, liquid-liquid, and gas-gas equilibria; they are compared with data for other systems taken from the literature. The results verify a previous hypothesis that there are continuous transitions between liquid-gas, liquid-liquid, and gas-gas critical phenomena in fluid binary mixtures at high pressures. The measurements are of interest for fluid extraction and supercritical fluid chromatography (SFC).