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Is there a constant value for $E$, in terms of $E=F/Q$, in units of $V/m$ or $N/C$, assuming the distance is negligible? (Virtually touching?)
What is the value of E in E=F/Q if the Q is the elementary charge; say an electron, and (again) the distance is virtually nil?
EDIT: What is the value of F for an electron or proton? In Newtons?
How many Newtons are there for a single Coulomb of charge? (Assuming, as always, negligible distance.... Or perhaps 1 meter, the SI unit for V/m....)
Are you essentially asking what happens to: $$E = \frac{e}{4\pi\epsilon_0r^2}$$ as $r \rightarrow 0$ ?
I'm assuming you're taking: $$F = \frac{Q_1Q_2}{4\pi\epsilon_0r^2}$$ with $Q_1 = e$.
In this formulation $E \rightarrow \infty$
However, things operating at this scale are a bit more complicated, and in reality the Uncertianty Princiapl and / or the Pauli Exclusion Principal would prevent this kind of thing from happening.
For a given problem with a given distance $E$ will be a constant value around an electron, until you get TOO close and other effects start to become noticeable.
In general, getting close to an arbitrary charge doesn't yield any meaningfully general constant value of $E$.
