The Mpemba effect and its inverse can be understood as a result of nonequilibrium thermodynamics. In polymers, changes of state are generally non-equilibrium processes. However, the Mpemba effect has been rarely reported in the crystallization of polymers. In the melt, polybutene-1 (PB-1) has the lowest critical cooling rate in polyolefins and tends to maintain its original structure and properties with thermal history. A nascent PB-1 sample was prepared by using metallocene catalysis at low temperature, and the crystallization behavior and crystalline structure of the PB-1 were characterized by DSC and WAXS. Experimentally, a clear Mpemba effect is observed not only in the crystallization of the nascent PB-1 melt in form II but also in form I obtained from the nascent PB-1 at low melting temperature. It is proposed that this is due to the differences in the chain conformational entropy in the lattice which influence conformational relaxation times. The entropy and the relaxation time can be predicted using the Adam-Gibbs equations, whereas non-equilibrium thermodynamics is required to describe the crystallization with the Mpemba effect.