The continuous packing of a flexible rod in two-dimensional cavities yields a countable set of interacting domains that resembles nonequilibrium cellular systems and belongs to a new class of lightweight material. However, the link between the length of the rod and the number of domains requires investigation, especially in the case of non-simply connected cavities, where the number of avoided regions emulates an effective topological temperature. In the present article we report the results of an experiment of injection of a single flexible rod into annular cavities in order to find the total length needed to insert a given number of loops (domains of one vertex). Using an exponential model to describe the experimental data we quite minutely analyze the initial conditions, the intermediary behavior, and the tight packing limit. This method allows the observation of a new fluctuation phenomenon associated with instabilities in the dynamic evolution of the packing process. Furthermore, the fractal dimension of the global pattern enters the discussion under a novel point of view. A comparison with the classical problems of the random close packing of disks and jammed disk packings is made.