Fractal objects are complex structures built with a simple procedure involving very little information. This has an obvious interest for living beings, because they are splendid examples of optimization to achieve the most efficient structure for a number of goals by means of the most economic way. The lung alveolar structure, the capillary network, and the structure of several parts of higher plant organization, such as ears, spikes, umbels, etc., are supposed to be fractals, and, in fact, mathematical functions based on fractal geometry algorithms can be developed to simulate them. However, the statement that a given biological structure is fractal should imply that the iterative process of its construction has a real biological meaning, i.e., that its construction in nature is achieved by means of a single genetic, enzymatic, or biophysical mechanism successively repeated; thus, such an iterative process should not be just an abstract mathematical tool to reproduce that object. This property has not been proven at present for any biological structure, because the mechanisms that build the objects mentioned above are unknown in detail. In this work, we present results that show that the glycogen molecule could be the first known real biological fractal structure.