Ytdyklly

The magnitude of the electrostatic force between two charges $Q$ and $q$ separated by a distance $r$ is given by $$F=frac{kqQ}{r^2}$$ but the minimum value of $r$ must be $10^{-15} rm m$. Therefore, my question is can the electrostatic force ever be infinite?

This

but the minimum value of $r$ must be $10^{-15} rm m$

sounds like you found a reference to the so-called "classical radius of the electron", possibly with some figures for the radii of atomic nuclei, but you did not fully understand what the former means.

The 'classical radius of the electron' $r_mathrm{cl}$ is the radius at which a spherical lump of charge would have an electrostatic self-energy equal to the rest energy $E_mathrm{rest} = m_e c^2$ of the electron. But the key word there is "would": the electron isn't a spherical lump of charge: as far as we can tell, it is a point particle with no internal structure that we've been able to detect ─ with a current experimental precision of the order of $10^{-18}:rm m$.

It is true, on the other hand, that when you're considering the electrostatic interactions between point particles at length scales shorter than about $10^{-10}:rm m$ (give or take, depending on what you're doing) you're going to need to change your framework from a classical viewpoint to one based on quantum mechanics, in which electrostatics remains mostly unchanged, but the whole mechanics itself (including the meanings of concepts like "trajectory", "distance" or "force") changes. Once you make that leap, the question of whether the electrostatic force can have infinite values becomes pretty much moot - but the singularity remains.

Physically, an infinite force is not possible. The fact that the simple electrostatic model (Coulomb's law)

$F=frac{kqQ}{r^2}$

suggests an infinite (or at least an unbounded) force between two point charges as they get closer and closer together tells us that this must be an approximate model which does not hold for very small $r$. Either point charges do not occur in nature, or the $r^{-2}$ model is replaced by something else for small enough $r$.

The classical physics equation $F = frac{kqQ}{r^2}$ has to be interpreted using quantum mechanics for sufficiently small length scales. So, its probably not appropriate to say that the force becomes infinite. A typical rule of thumb for the smallest length scale for which this applies is the Compton wavelength $lambda = frac{h}{mc}$. Here $h$ is Plank's constant, $c$ the speed of light and $m$ the particle mass. For an electron, this comes out to $2.4times 10^{-12}$m, but for a proton, it would be smaller.