In all physical theories, any finite system has a finite number of distinguishable states. So it is not infinitely informationally dense, especially when working with discrete bits of information.
Not to mention, the finer the distinctions between two states of a system, the more energy you need to distinguish them. So, the less impact these differences can have, unless the system is extraordinarily energetic (and even then, you end up in fundamental limits of energy per volume, like the Schwarzschild radius).
So again, there is no sense in which a finite part of the universe is universally dense.
Even worse for your argument, all currently known laws of physics use computable functions (the randomness in QM notwithstanding). So, by definition, all known laws of physics can be simulated by an ideal Turing machine (again, give or take some randomness in QM, depending on the interpretation you chose to believe in and on how you chose to simulate the QM system).
i specifically said with a /discrete/ conception , geometry is infinitely dense (of discrete states)