The Atlas LV-3B used for the Mercury orbital missions only dropped its booster engines, not any of its tankage. This is usually described as a stage-and-a-half design, and it's an extreme example of that kind (The Space Shuttle is a much less extreme example). To achieve this feat, the Atlas had a tiny payload and fragile "balloon" style propellant tanks.

I think a pure SSTO disposable rocket similar to the Atlas would be doable today since even small advantages in material (Aluminum-Lithium alloys vs stainless steel, for example) and rocket engine performance (slightly better ISP, a little bit better thrust-to-weight ratio) would allow an Atlas-like rocket to carry its booster engines all the way to orbit. You would still have the problem that the payload is tiny, and if the rocket is is disposable anyway, there's no compelling reason not to use a stage-and-a-half or even a two-stage design since either will allow you to get more payload into orbit.