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Imagine. Our solar system. Our sun. Then earth and moon orbiting it. And you have "powers" to create any planet you want, any size, any density, any weight and any velocity. Would it be possible for you (using all knowledge of earth), to create a natural satellite to moon? Whose trajectory would be almost circular/ellipsoidal?

Question actually goes only as follows: Can moon of moon (actual moon) of moon (earth) of sun exist?

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This is possible. For example, this post provides a good explanation of the math involved.

The key point is whether an object lies in the planet's Hill sphere or the moon's Hill sphere:

Can the Moon have a moon?

Yes, the Moon could have a sub-satellite. If we look at a system of the Earth, Moon, and a sub-satellite, the same idea as above applies. The Moon has its own Hill sphere with a radius of 60,000 km (1/6th of the distance between the Earth and Moon) where a sub-satellite could exist. If an object lies outside the Moon's Hill sphere, it will orbit Earth instead of the Moon. The only problem is that the sub-satellite cannot stay in orbit around the Moon indefinitely because of tides.

As long as the gravitational attraction of a satellite is consistently stronger than the body the satellite orbits, a third body will orbit the satellite.

There are obviously limits to how small you can go with this, as gravity is a relatively weak force at small scales. At some level, either the Hill sphere is smaller than the object itself or other forces dwarf gravity's impact. For example, a person's gravitational field is too weak to attract its own satelite; the Earth wins that one every time.

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Now that's what I call explanation. Thank you. I will mark as correct answer as soon as possible. –  Rik Telner Feb 3 at 17:34
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"The only problem is that the sub-satellite cannot stay in orbit around the Moon indefinitely because of tides." and also the next paragraph (in the linked text) suggest, to me, that the answer is: "No, it is not possible"! –  Glen The Udderboat Feb 3 at 17:35
    
@GlenTheUdderboat Well, theoretically, nothing can orbit indefinitely in the real universe -- stars collapse, impacts knock things off-course, and so on. There's no reason, though, that you couldn't make something orbit pretty stably for, say, the entire expected lifespan of the moon (i.e., until the sun swallows it). –  Ed Cottrell Feb 3 at 17:37
    
Relax. I've read the answer. –  Rik Telner Feb 3 at 17:38
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If things are in close, you have the three-body problem (four, given the sun - but it is distant). Three-body orbits are not uncommon in the universe. Orbit evolution prediction is fundamentally impossible, though abundant special solutions exist,

http://en.wikipedia.org/wiki/Three-body_problem
http://news.sciencemag.org/physics/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problem

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Remember part in topic post saying "any size, any weight"? It can weigh for me 1kN and fly 1km from surface. As long, as it stays within Moon's orbit for considerable amount of time (200 years?). –  Rik Telner Feb 3 at 18:25
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