Which of these organelles functions as the digestive system of the cell?
Lysosomes are spherical vesicles enveloped by a single membrane. 
Lysosomes are regarded as the digestive tract of the cell, since they are actively involved in digestion of cellular substances-namely proteins, lipids, carbohydrates and nucleic acids. 
Lysosomal enzymes are categorized as hydrolases.
The lysosomal enzymes are responsible for maintaining the cellular compounds in a dynamic state, by their degradation and recycling. 
The degraded products leave the lysosomes usually by diffusion, for reutilization by the cell.
Sometimes, however, certain residual products, rich in lipids and proteins, collectively known as Iipofuscin accumulate in the cell. 
Lipofuscin is the age pigment or wear and tear pigment which has been implicated in aging process.

Key Concept:
Lysosomes are regarded as the digestive tract of the cell, since they are actively involved in digestion of cellular substances-namely proteins, lipids, carbohydrates and nucleic acids. 
Ref: Textbook of Biochemistry, Satyanarayana
Mitochondria
Rough endoplasmic reticulum
Golgi apparatus
Lysosomes