Can we ever escape Earth's gravity without using chemical propulsion?

Of course. Space elevator were suggested more than a century ago. Arthur Clark made the idea popular in the sixties.

Perhaps a railgun plus a skyhook? Railgun up the side of a mountain on the equator and an orbiting station with a capture mechanism that is in a sense the strongest bungee cord imaginable. 17k mph is a lot of speed to make a grab. Exiting the railgun at mach 3-4 at 30,000 feet has engineering issues and still has accelerate above mach 20 for a stable orbit.

Laser propulsion is an interesting / wacky idea. Don't know if it would actually work, but the core concept of keeping the energy source on the ground is an interesting one at least.

https://en.m.wikipedia.org/wiki/Laser_propulsion

Giant railguns/coilguns for shooting cargo and/or passenger modules into space? This has been done in science fiction at least.

Is NERVA TSTO possible, strictly in technical sense? Yeah why not?

Neal Stephenson has a lot to offer in the Seven Eves