Imagine I perform a random sequential adsorption (RSA) simulation for circles or discs of some radius $r \leq 1$ in $[0, 1]^2$ (I am open to changing this geometry to the unit circle). As a function of $r$, what is the best known lowerbound for the number of discs I can place on the surface s.t. all discs are guaranteed to fit? In other words, if I simulate RSA until I place all $N$ discs on the bounded surface, for some value of $r$, what is the largest known $N$ s.t. my simulation guaranteed to halt?
If you can't fit another disc with radius $r$ in, then it means that every point is within $2r$ of one of the current centers. So, you are asking how efficient a covering by discs can be. Asymptotically, the most efficient covering is the hexagonal one, with efficiency $\frac{3\sqrt{3}}{2\pi}$. So, in a square or disc of area $A$, you can place at least $A/((6\sqrt{3} + o(1)) r^2)$ discs of radius $r$. The geometry of the region doesn't affect the asymptotics, but you can't simultaneously let the region vary as $r$ does, or that would allow you to consider a union of discs of radius $2r$, which can be covered by discs of radius $2r$ with perfect efficiency. 

