Studying the growth of 1D structures formed by the self-assembly of crystalline-coil block copolymers in solution at elevated temperatures is a challenging task. Like most 1D fibril structures, they fragment and dissolve when the solution is heated, creating a mixture of surviving crystallites and free polymer chains. However, unlike protein fibrils, no new nuclei are formed upon cooling and only the surviving crystallites regrow. Here, we report how trapping these crystallites at elevated temperatures allowed us to study their growth kinetics at different annealing times and for different amounts of unimer added. We developed a model describing the growth kinetics of these crystallites that accounts for fragmentation accompanying the 1D growth process. We show that the growth kinetics follow a stretched exponential law that may be due to polymer fractionation. In addition, by evaluating the micelle growth rate as a function of the concentration of unimer present in solution, we could conclude that the micelle growth occurred in the mononucleation regime.