Single Stages That Could Go All The Way

I've read that there are stages of rockets that have actually flown
that had a high enough ISP and mass ratio they could have gone to
orbit if they did not need to carry a payload. In other words, if
someone had fired off the first stage of the Saturn 5, it could have
made orbit all by itself except it was carrying the rest of the rocket.

Is this true? Can someone offer some examples? And what practical
problems (like bringing along an aerodynamic shield for the nose) might
one encounter in real life.

Note, I'm not asking about hypothetical SSTO rockets, but real flown
stages.

-Charles Talleyrand