What follows is from Professor Barton Zwiebach of MIT:
The only problem with such a large field is an infinite self-energy of the point particle limit. This problem is not really there in QED, as any infinite self energy is renormalized away. There is actually an effective version of Maxwell’s equations, called nonlinear electrodynamics in which point charges have finite self energy, and electric fields cannot exceed a limit (just like the speed of objects is $< c$). This is also true in classical string theory. It is explained in Chapter 20 of my book on string theory."
What does it mean that the energy is renormalized?