HOTOL

Thumblegudget talks HOTOL [h/t ToughSf]. This was a British proposal to get around the Smithsonian Problem.

The issue here is that although Superheavy and Starship exist (sort-of), they didn't exist in the 1980s. Multiple-stage rockets were expensive on account the early stages weren't retrieved. Proposals back then noted that Earth actually does allow for retrieval of the first stages if these stages are air(=oxygen)-breathing. Which we don't get over any other planet in this system.

But then they'd be: jet, ramjet, scramjet, orbit. Assemble everything from a comprehensive museum of aeronatics! Hence the Lightcraft, among others. Not only is this a recipe for error at any point in the process, but after each engine is done we don't need the engine. So you're bringing a lot of extra mass up there. Retrieve each piece now, or retrieve it all after the mission is over...

This isn't my field so all I've done here is summarise Thumblegudget, whose youtube I recommend.

Thumblegudget explains some core principles. He starts with the heat-engine on the classic thermodynamic cycle: suck, squeeze, bang, blow. After "blow", the classic solution reheats the high-pressure gas in the main-combustion chamber / "afterburner". There's an Expander Cycle which is more efficient, but those aren't good for thrust so tend to the second-stage (like the Starship).

HOTOL - says Thumblegudget - is a modification of the Expander Cycle, which can do high-thrust down below and high-efficiency up above. Only one engine needed. Overall efficiency suffers - but it's all much more efficient than carrying a second engine you only want half the time.

This was made possible by compressed liquid hydrogen at 15 K; the hydrogen that isn't burned (yet) can be used as a coolant in the earlier parts of the cycle. When the Brits asked, what about the water-vapour (with a U) in our air that will freeze up inside the machine; this project was, er, cancelled. Later researchers who carried on this design introduced antifreezes.