Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$: 


Under "no overlap" constraints, I sequentially deposit circles of radii $r\_c$ on this surface, where the center-point of each circle is allowed real number coordinates and is chosen with uniform probability. To address concerns related to edge conditions, only the center-point of each surface needs to be on the bounded plane. If a set of random coordinates imply an overlap with one or more previously deposited circles, a new set of random coordinates is chosen. Under no conditions may any of the previously placed circles be rearranged. 


Under these conditions, approximately how many circles, $N$, should I be able to pack onto the plane? What average-case maximum packing density is achievable? 



Packing density of randomly deposited circles on a plane
The model you describe seems to fall under what's called **"Random Sequential Addition"** or **"Random Sequential Adsorption"** in the literature; it's viewed as a higher dimensional analogue of the car parking problem. An early review in the physics literature on this type of model by J W Evans is [here](https://doi.org/10.1103/RevModPhys.65.1281 "J. W. Evans, Random and cooperative sequential adsorption, Rev. Mod. Phys. 65, 1281"). From this review, I found [a paper by Einar L. Hinrichsen, Jens Feder and Torstein Jøssang](https://doi.org/10.1007/BF01011908 "Hinrichsen, E.L., Feder, J. & Jøssang, T. Geometry of random sequential adsorption. J Stat Phys 44, 793–827 (1986)") which discusses continuum RSA of disks in the plane. Their simulations yield that for large A,B the fraction of space filled in the jammed state is around $\theta\_J=0.5472\pm0.002$, from which you should be able to extract the answers to your questions.


There's a newer review [here](https://doi.org/10.1088/0953-8984/19/6/065124 "A Cadilhe et al 2007 J. Phys.: Condens. Matter 19 065124") by A Cadilhe, N A M Araújo and Vladimir Privman.


There's a ton of more recent work, but this should give you a place to start looking.


