The scope of this work is to investigate polymer crystallisation involving thickening growth by means of a kinetic Monte Carlo simulation. The simulation combines the lateral growth mechanism of the Sadler/Gilmer model with Hikosaka's 'chain sliding diffusion' mechanism for thickening growth. As a result of this coupling of lateral and thickening growth, we obtain wedge-shaped crystals if the sliding friction is not too high ('mobile regime') and lamellar crystals for high friction. In the mobile regime the thickening rate increases with undercooling. Thickening also increases the lateral growth rate, especially at low undercooling, since it lowers the entropic and enthalpic barriers. As a result, the activation energy of both processes takes a similar value, in agreement: with experiment. For high friction, the thickening rate increases with temperature up to a maximum close to the melting point, in agreement with the observations of thickening of lamellar polymer crystals. This simple model hence captures the essence of lateral and thickening growth behaviour in polymer crystallisation.