Assuming site saturation and diffusion-controlled growth, the volume of the growing body is proportional to t^3 in 3D growth and proportional to t^2 in 2D growth (where t is time). In the parlance of Kolmogorov-Johnson-Mehl-Avrami kinetics, this leads to an Avrami exponent of 3 for 3D growth and 2 for 2D growth.

Please precise the question. What are the conditions? How many phases are present in the system? Do you grow layers by deposition techniques? Your open question can be answered in many many ways. Good luck.

Layer-by-layer growth is called epitaxy if the growth is of a single crystal on a single crystal substrate. This is how molecular beam epitaxy (MBE) works (in ultra-high vacuum, UHV). The same effect can be achieved at much poorer vacuums by chemical vapour deposition (CVD). CVD is often configured for epitaxial growth.

The same layer-by-layer epitaxy is regularly used in the semiconductor industry to anneal silicon (and other) devices after ion implantation. High fluence ion implantation can amorphise the crystal, and annealing can regrow the material epitaxially (essential to make devices!).

On the other hand, if for example one deposits CVD silicon on glass (this will be amorphous if deposited below say 400C) and anneals, there will not be epitaxy since there is no underlying single crystal seed (substrate is glass!). In this case crystalline nucleation will occur randomly throughout the body of the CVD layer. This is 3D nucleation.

Assuming site saturation and diffusion-controlled growth, the volume of the growing body is proportional to t^3 in 3D growth and proportional to t^2 in 2D growth (where t is time). In the parlance of Kolmogorov-Johnson-Mehl-Avrami kinetics, this leads to an Avrami exponent of 3 for 3D growth and 2 for 2D growth.