A star collapses in stages depending on it’s initial mass:
A Stellar Mass on par with our Sun will collapse to a white dwarf.
A little bigger star will collapse to a neutron star: just a smidgen bigger, and you collapse to a quark star.
At about 3 solar masses, you infinitely collapse to a quantum singularity…a Black Hole.
in Planck units, what magical density does infinite collapse take place, and how does that relate to the Planck density?
I don’t believe the answer to your question is exactly known for the evolution of a stellar black hole, mainly because the value of the Tolman-Oppenheimer-Volkoff limit, which gives the maximum mass for neutron stars, isn’t known exactly.
Generally, though, if you compress any amount of mass into a volume smaller than (or equal to) a sphere whose radius is just the Schwarzschild radius for that mass, you will get a black hole as a result, corresponding to a density way, way lower than the Planck density.