My understanding of an electron is that it's surrounded by virtual particles. Does it annihilate with a virtual positron, leaving a virtual electron which then goes on to repeat the process?
These kinds of questions basically come from two common misconceptions:
1) Confusing classical with quantum It doesn't make much sense to talk about what an electron does, not in the classical sense. Fundamentally, there exists an electron field which permeates all space. Furthermore, it is a quantum field whose excitations are quantized. That's what we usually call the electron. The same goes for all fundamental particles. According to fundamental principles of quantum mechanics, it does not make sense to talk about what the particles do, unless we're talking about outcomes of some measurement. Roughly speaking, if we measure a system at the state A and later at state B, we could say that it evolved from A to B in every possible way 'at the same time'.
2) Confusing Feynman diagrams with real processes
Everyone knows what a Feynman diagram looks like and even laymen can understand its building blocks and say what went it, what interacted and what came out. But they are simply a pictorial representation of the perturbation series expansion in quantum field theory. In other words, it's a series of terms which get (if we're lucky) more and more precise with each additional term. One diagram represents one of those terms and it translates to an integral that we have to calculate. Virtual particles are those that don't have external lines in a diagram, only internal. They don't have to conserve energy and momentum and they are never directly observed. In a sense, if you take every diagram (infinite amount!) you could tell a story, because all of them happen. If two particles go in and two go out, everything that can happen happens in between.
It's hard to talk about these things clearly because QM is not easily described by everyday language. Furthermore, the formalism of QFT forces you, in a way, to think about the initial situation and the final situation. What happens in between is what we calculate but it's hopeless to try and interpret each line of the calculation as something physical and tangible.
I hope I didn't miss the point completely :)