Could a star revolve around a planet? They actually do, but let's set Relativity aside for a moment. And the whole barycenter thing too. Here's the scenario:

Somewhere in the backwaters of a galaxy there is this huge planet. It is NOT a failed star. It could be considered a gas giant, except most of the gas is nitrogen, with traces of oxygen. The planet has a nucleus with a rocky surface covered in part by water. There's life on the planet, maybe even intelligent life. Orbiting this planet is a dwarf star. It is sufficiently close to the planet to provide energy and warmth, and not close enough to incinerate it. Its orbital speed is sufficient to warm the equatorial parts of the planet without singing them when it passes over them and to come back to them before they freeze over. Or maybe there are two stars of slightly different sizes, traveling along different orbits. 

Is this at all possible?
And the whole barycenter thing too.


You're throwing out the definition of orbit at this point, and moving into a different realm. If you throw out barycenter, you have to leave me with my frames of reference.

A star can definitely orbit a planet, if one such as this did in fact exist. You are assuming it is simply a planet-star system, which makes things easier. If you're on the planet, the star is orbiting you, if you're on the star (ouch), the planet is orbiting you.

Once they are of comparable size, the barycenter thing has to be taken into account, you can't just throw out this crucial part of the physics, even in a thought experiment. If you want that part out, the planet must be much bigger than the star.