Probably the closest actual case was the original Atlas, which reached
orbital velocity with the entire fuel tank but without its booster
engines.

For true SSTO with nothing dropping off, the best bet would be the
Delta IV core stage, because of its long burn time and extremely high
ISP. It is the only rocket that can take off with hydrogen fuel alone
(no strap-on boosters).

Without the second stge, the core stage could probably get into orbit
assuming it didn't exceed aerodynamic pressure limits on the way up:
Gross Mass: 226,400 kg. Empty Mass: 26,760 kg. Thrust (vac): 337,807
kgf. Isp: 420 sec. Burn time: 249 sec. Isp(sl): 365 sec. .