Fluctuation conductivity is experimentally studied in the genuine critical region near the superconducting transition of YBa2Cu3O7-delta, YBa2Cu2.985Fe O-0.015(7-delta) and Y0.95Ca0.05Ba2Cu3O7-delta single crystal samples. Two fluctuation regimes where the electrical conductivity diverges as a power-law of the reduced temperature were systematically observed. In the first regime, farther from the critical temperature T-c, the transition behaves as predicted by the thermodynamics of the three dimensional-XY (3D-XY) universality class characteristic of a second-order phase transition. In the asymptotic regime closer to Tc a power-law regime characterized by a much smaller exponent is observed. The smallest value ever reported for the fluctuation conductivity exponent in the high-T-c superconductors is obtained for the Fe- and Ca-doped systems. We suggest that the regime beyond 3D-XY is a crossover towards a weakly first-order transition induced by internal magnetic excitations. (C) 2017 Elsevier B.V. All rights reserved.