Here is a simulation of the Falcon 9 first stage as a SSTO (Single Stage To Orbit) vehicle. There would be no practical reason for SpaceX to demonstrate this. However, I wanted to confirm the speculation that it could at least be theoretically possible. I try to keep the stress load on the rocket under 4.5Gs at all times by shutting down pairs of symmetric engines. This is a pretty aggressive flight profile, but I’m having trouble finding anything else that works. This barely makes it to LEO with a final orbit of around 180km x 170km.
The simulation software I wrote can be found on GitHub here:
If you just want to run the simulation here is a link to the program: