The primary limitation on the size to which a single cell can grow is a mathematical principle called the surface to volume ratio. As the size of a three-dimensional object grows, its volume increases more rapidly than its surface does, which causes metabolic problems for cells. Additionally, the amount of cytoplasm the nucleus can contain and the structural limitations on the cell prevent them from being larger as well.
Cells are discrete metabolic units. They must be able to take resources in and expel waste and energy. The only place a cell can do this is along the thin, skin-like membrane surrounding it. As the volume of the cell increases in size, it must acquire and expel more substances; however, because the volume grows more quickly than the surface area, there is a limit to the amount of diffusion that can take place into or out of a cell.
The nucleus of a cell is essentially a small sphere within a larger sphere. Because the nucleus must become larger to control a larger cell, the nucleus is also susceptible to the problem of surface to volume ratio. This limits the size of the nucleus, which in turn, limits the size of the entire cell.
While the outer membrane of a cell protects the cell well on a microscopic level, large cells would require exceptionally thick membranes. As these membranes thicken enough to hold larger cells, they suffer from decreased permeability.