Cell's ability to proliferate constitutes one of the most defining features of life. The proliferation occurs through a succession of events; the cell cycle, whereby the cell grows and divides. In this paper, focus is made on the growth step and we deal specifically with Saccharomyces cerevisiae yeast that reproduces by budding. For this, we develop a theoretical model to predict the growth powered by the turgor pressure. This cell is herein considered as a thin-walled structure with almost axisymmetrical shape. Due to its soft nature, the large deformation range is a priori assumed through a finite growth modeling framework. The used kinematics is based on the multiplicative decomposition of the deformation gradient into an elastically reversible part and a growth part. Constitutive equations are proposed where use is made of hyperelasticity together with a local evolution equation, this latter to describe the way growth takes place. In particular, two essential parameters are involved: a stress-like threshold, and a characteristic time. The developed model is extended to a shell approach as well. In a finite element context, representative numerical simulations examining stress-dependent growth are given and a parametric study is conducted to show the sensitivity with respect to the above mentioned parameters. Finally, a suggestion for natural contractile ring modeling closes this study.