My question revolves around how dense could you make a human/animal body before it breaks under its own weight or before you couldn't stand up.
That's not what you have to worry about.
If your mass remained the same, but the density was altered, than you would shrink. I am assuming this to be the case, as you say that the magic system allows for an adjustment of density, but not mass so I'm assuming that the law of conservation of mass is unaltered in this case. This is where I invoke the square-cube law, but the thing is that we're assuming the mass remained the same, so the only change here is relative surface areas. (We're going to assume that you adjust the density ratios on the fly to keep the proportions intact in the smaller form.) This is bad, because the surface areas shrink, limiting the rate of exchange for important things, like gas exchange in the lungs.
That is to say, when density increases, you'll have trouble breathing. Doubling the density of a human being would lower the surface area by 1/4, given that you're essentially cutting the size in half. And that isn't just the lungs that's affected, it's anything that needs exchanges across a membrane all across the body, effectively cutting everything to a quarter. Which is more or less deadly. Honestly, if you want to be science-based and biological about this, increasing density without improving mass is a death sentence.
Increasing density while increasing mass isn't that much better, by the way, because then your body needs to upkeep that mass as well and you run into the same problem. Humans operate at the current density because it works well for us, adjusting the density by a major factor would be very, very problematic. Unless you just want to use magic to ignore all these problems.