Nuclear pulse propulsion, aka Orion. There's little doubt it would work. The limits are not technological, but the fact that boosting an Orion means you're detonating a lot of atomic bombs in the atmosphere.
If you'll accept launches from other bodies the mass driver is the best answer. You can make the track as long as you like. Many years ago on TV I saw one that could boost a projectile to 100mph in six feet. If my understanding of the math is right that means you could easily scale it up to lunar escape velocity with a 3.2 mile track. Punching through the atmosphere is quite another matter, while a track a few hundred miles long could safely boost an astronaut to Earth escape velocity he wouldn't survive the passage through the atmosphere and the capsule would fall back due to drag anyway.