Strictly speaking, the energy gained by a charged particle along any path is the integral of the dot product between the electric field and the line element alo

Strictly speaking, the energy gained by a charged particle along any path is the integral of the dot product between the electric field and the line element along the path (edit: times the charge). In the absence of changing magnetic fields, this is always the difference between the voltage at the starting and ending points. What I was getting at is that there's only so much energy a field can put into a particle. Either the voltage will drop because the machine can't "keep up," or the particle will reach a terminal velocity where the force applied by the field is insufficient to accelerate the particle any more. But, barring those two things, the particle will continue to accelerate.