Technically this is just a subtype of Verne gun. Which fun fact kinda existed. Back in the 60s HARP fucked around with a power charge based one down in Baja to achieve sub orbital space flight for a satellite.

Also Sadam Husein wanted to create one so do with that what you will.

To put something in orbit, it has to go sideways very quickly. It has to rotate around earth, such that the free-fall causes the curvature of the circle. For Low Earth Orbit, that's 7 km/s. You have to get it to that speed, just "jumping" isn't enough. You'd need some kind of railgun or catapult.

You definitely wouldn't want to do this all the way to geosynchronous orbit. Just getting it to the edge of space is already ridiculous to the point where it has me questioning how much pressure and heat the stone at the bottom would reach, and therefore how stable it would be.

And for a super earth, getting out of the soupy atmosphere is a challenge in of itself, so getting rid of that challenge would already be an incredible head start. From there you'd just need engines powerful enough to get you up to speed before hitting the ground.

Like one thing is how tall can you get before the base encircles the planet (where trying to add more layers just makes the planet bigger and requires bringing in outside material, which means your geosync orbit gets farther).

Gut vibe tells me that probably wouldn't be a problem, as the atmosphere on any given planet, even a super earth, is only about as thin as the skin of an apple relatively speaking. And that's all you'd need to beat here.

Could a large enough pyramid give the planet a wobble?

Absolutely. Though again gut vibe tells me it would probably only be a wobble of a few millimeters, nowhere near enough for anybody to feel it.

Assuming you had a perfectly strong material that could handle it, is it possible to build a tower to a geosynchronous orbit or will it keep moving away as you add mass to the tower?

I think at a certain point you'd be far enough up that you could reasonably just build a space elevator on top of the pyramid out of normal-ish materials like steel. The farther up you start the less of a foot you have in the gravity well, and the less distance your steel needs to support. At that point it would maybe be worth it do build a counterweight and go to geosynchronous orbit.

Another thing to keep in mind, if some civilization was crazy enough to do this, hopefully they'd be smart enough to do it around their equator to reduce the amount of pyramid of doom they'd need to build. But that would probably also mean bulldozing lots of countries and mass migrations.

It definitely would, but I'm guessing you'd run into the issue of stability far before that.

I'm also guessing that the ratio of atmospheric extension to terrain extension would be on the side of terrain extension. Gravity is a rather weak force in comparison to the other forces of the universe.

This would be a fantastic xkcd "What if?" question if it isn't already.