Many theoretical physicists consider the Planck density to be the maximum where density has a measurability and thus meaning within general relativity and quantum mechanics theory. (These quantities are already pushing well into the realm of Heisenberg limitations measurement fuzziness, and in the most extreme of general relativistic effects.) From the http://www.britannica.com/science/cosmology-astronomy/Relativistic-cosmologies#toc27602 . By definition, of the Planck volume, and the Planck mass: "An object of such mass would be a quantum black hole, with an event horizon close to both its own Compton length (distance over which a particle is quantum mechanically 'fuzzy') and the size of the cosmic horizon at the Planck time.". So, packing any more matter in would not yield any measurable or feature-imaginable effects, unless we had a quantum gravity theory to describe it. Simply reckoned, it might explode, as in another BIG Bang. In fact, the concepts are taken from early after the BIG Bang, not long after the initial shock waves, and the speed-of-light has settled to a constant, related to the density, and the general relativity and other laws of physics are just starting to govern. http://abyss.uoregon.edu/~js/images/planck_era.gif
It is the echo of the BIG Bang - the background radiation from the superluminal,(faster-than-light, I like to imagine) post Bang events - which persist like a cosmic Krakatoa (that bang circled the earth only four times), but with an budding universe so dense, and the Hubble Telescope, and now the experience of measuring gravitational waves, we have an incredible model which may soon be more refined.
So, like the others they indicate the extremes, shortest meaningful, measurable time, smallest black hole, and then, the maximum density. These quantities are very rough and can be used for sanity checks on models and orders of magnitude checks, as they arose from a dimensional analysis of quantities as they were first resolved.