We study the breakdown of the magnetization plateau at the magnetization M = M_S/3 (M_S is the saturation magnetization) of the S = 1/2 anisotropic spin chain with ferromagnetic-ferromagnetic-antiferromagnetic interactions. We consider the model with the isotropic ferromagnetic (trimer) coupling J_F, and anisotropic antiferromagnetic coupling (J_x = J_y = J_AF and J_z = J_AF). For the limit of large J_F/J_AF, the model is equivalent to the S = 3/2 XXZ chain with the exchange anisotropy . There is a phase transition between the plateau (small-) and the no-plateau (large-) regions. This phase transition is of the Berezinskii-Kosterlitz-Thouless type, and we determine the phase boundary from the numerical diagonalization data. For = 1, in particular, the phase transition between the plateau and the no-plateau regions occurs at the point _c = 15.4.