I'm curious, but does anyone know how many different forms of staged combustion exist or have been thought up? I can think of at least three off the top of my head.
Twin-preburners (one oxygen rich, one fuel-rich)
Examples: RD-170 family
Single pre-burner & turbopump (runs oxygen-rich or fuel-rich)
NK-33, RD-162, RD-0120
Full-flow staged combustion
Examples??
Does the SSME count as a unique staged combustion cycle due to its resonance chambers?
You're confusing a few things.
1) About staged combustion the best way to think about it is the Russian nomenclature. They speak about how fuel and oxidizer arrive at the chamber's injectors. Thus, a gas generator is liquid/liquid, a staged combustion is gas/liquid and a full flow is gas gas. The only gas/gas engine that I know about, was the failed RD-270 engine. And there was the Integrated Powerhead Demonstrator. But I don't think they even had a thrust chamber for it.
2) RD-170/1/1M family has two identical oxidizer rich gas generators. That's why making the RD-180 required so little changes. When they developed it they didn't felt confident on making a single gas generator of that side. But both gas generators are oxidizer rich and only the oxidizer arrives as a gas to the chamber. The fuel is injected in liquid form.
3) The H2 and CH4 engines might extract some extra energy by doing something like an expander, but it's usually to avoid cavitation or such. In the RS-25D case the H2 LP turbine I think it does use expander power, but I'm not sure. I know that the only use of the LP turbines is to avoid cavitation.
4) You have to consider the expander cycle, for example. You have both closed cycle (like the RL-10/Vinci/RD-0146), that inject gas fuel and liquid oxidizer. And you also have open cycle expander (like the LE-5A/B and LE-X), where the expanded fuel is dumped and the fuel that does arrives to the injectors is in liquid form.
5) strangequark did teased about a new cycle for the RL10 replacement program.
6) You also have to consider the pressure fed and piston pump fed engines, you might consider them different cycles.