My question is about real physics or what is still possible given humanity's current understanding of how the universe works. No magic permitted, no bullshit with scientific terms and expressions mixed in.
Technologies don't necessarily have to exist but need to be possible or at least plausible to the extend where they can't be ruled out using our current understanding of physics.
Let's say you live on a planet with the same density as earth but with twice its radius. This means double Earth's gravity on the surface and double Earth's escape velocity. You want to explore a different planet. How do you escape that planet's gravity?
The technological civilization on the planet has a very strong urge to get a probe off the planet. Funding is granted, they have our current technology except for space travel (which they never did), and they got 100 years of time until the probe has to leave.
Such a planet has twice Earth's gravitational force on the surface. Because this has been doubted when I mentioned it in the comments, it wrote the concept down:
The image is scaled down so the website loads faster. High resolution is available.
Now, this is the rough concept which fits on a page and probably would satisfy physics teachers in school. If you don't believe it, I can write you a scientific proof but that will be several pages long and it preferable shouldn't be after 3 in the morning where I live.
Note: I forgot to mention that those formulas are valid for points above the surface of spherical objects even though they presume the mass to be a single point in the center of that object. That's a generally known fact, though, and proving it is really simple. I can prove that or you can just look it up on Wikipedia or something.