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Earth wouldn't coalesce without the sun. Thus no geothermal.
Nuclear materials were formed in supernovas. Can't have that without a star.
Moons tides wouldn't exist if Earth didn't have an orbit of its own. Again, sun needed.
Hydro you already figured out.
I guess it depends on how far back you want to go. The sun wouldn't exist without gravity.
Well, seeing as how the question is directly asking "is there any energy you can't eventually trace back to the sun", you shouldn't ever stop going back unless you can reach a definitively non sun answer
(Gravity arguably works - probably the nuclear attraction force as well, electromagnetic I believe)
Yeah, although I'm not sure if I'd consider the four fundamental forces as "energy", exactly.
When most people ask the question, they mean in the context of the current environment, where the Earth, moon, etc. are already formed.