Suppose one places unit cuboids of dimension $d$ with min-corners uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden. At some point, the region is "saturated," without room for more cuboids to fit. For example, for $d=2$ squares in $[0,10]^2$, it appears that about $63$ squares on average will fit (whereas $n^d=100$ could fit):
So here the packing density is about $63$%.
Q. Is the expected number of cuboids known, as a function of $n$ and of $d$? Is the growth rate in $n$ known for fixed $d$?
Update. Benjamin Dickman identified the $d=1$ version as the Rényi parking problem, where it is known that, for large $n$, the packing density approaches $m \approx 0.7476$, i.e., the expected number of cars(intervals) is $m n$. It is conjectured that in $d=2$, the expected packing density approaches $m^2 \approx 0.559$.
Yoav Kallus points to a related paper that uses the apt phrase sequential random packing for the process.