Have three or more celestial bodies rotating each other in a stable manner ever been observed, or it is only a theoretical problem?
Answering in the spirit of the question, I think he's asking if there is ever a chaotic 3 body system that's long term stable, or, to put it another way, a 3 body system without a standard hierarchy where it's stable.
The answer is no. 3 body systems without Heirarchy are never stable for very long. They can certainly exist for a while, but they aren't long term stable.
What do we mean by Hierarchy
The Sun-Earth-Moon is an example of a long-term-stable 3 body system in heirarchy. It will remain stable for billions of years. The Sun is pretty much in the center. The Earth-Moon system orbits the sun and the Moon also orbits the Earth. In a sense, the Moon orbits both the Sun and the Earth. The Moon can orbit the Earth because it's inside the stable part of Earth's Hill Sphere.
Hierarchies beyond 3 might exist, such as Sun, Planet, Moon, Moon's satellite but there are no natural satellites of Moons in our solar system.
Obviously, multiple planets around a comparatively massive Central star can be long term stable, and just for clarity, lets call a system stable if it lasts a billion years. Mercury, Venus, Earth, Mars, Juptier, Saturn, Uranus, Neptune are obviously stable and mathematical models say that they should remain stable for at least the next 3.5 billion years, after which, there's a chance Mercury could become chaotic and fly out of it's orbit, flying into the sun, crashing into another planet or perhaps leaving the solar system.
Any of these orbital systems, for any kind of long term prediction of where the planets will be in a billion years, requires n-body calculations which are very complex and inexact, but they can be calculated within a range of error, which makes it possible to say that a system is stable, for a couple billion years anyway. You can't, for example, ignore the other planets and calculate where the Earth will be in 50 million years. The planets affect Earth's orbit but they don't destabilize it. This is called perturbation theory.
And the final example, binary star system with a planet orbiting the two stars.
This XKCD discusses the binary star system, and from same:
A planet orbiting two stars can't get too close to them or its orbit becomes unstable. If it gets too close, the irregular tugging from the gravity of the two stars as they orbit will eventually cause the planet to crash into one of them or get flung out of the system.
For a system with two similar-sized stars, this "critical radius" is around six times the distance between the two stars.
What this means is that any 3 body system has to follow the rules if it's going to be stable. You can have several planets orbiting a central star, and those planets will perturb each other, sometimes casting a planet out but you can also have stability.
You can have Sun, Planet, Moon, but the Moon must be inside the stable region of the Hill Sphere.
And you can have one object orbiting two central objects, like Tatooine, or, like the 4 satellites that orbit Pluto-Charon.
But anything outside those rules is an unstable 3 body or n-body system and it may take some time, but it's ultimately unstable and unlikely to last anywhere close to the billion(s) of years or so that stable orbits tend to last.
So a star with two planets who's orbits cross each other (note, Pluto and Neptune don't actually cross, they just appear to cross when drawn in 2 dimensions). Their orbits don't come close to crossing.
Horeshoe orbits are 3 body orbits that might last a fairly long time, but I don't believe they are truly long-term stable (if anyone can verify that, please do), but intuitively, I don't see how horeshoe orbits could last very long.
When you have a 3 body system where there's no structured orbit and all 3 bodies kind of dance around each other. That system isn't stable.
If properly balanced, a 3 body system outside the rules of stability might last for hundreds or thousands or orbits, but in celestial mechanical time that's not stable.
Mathematically some 3 body orbits have been worked out, but they are stable in the same way that a pencil is stable if you balance it perfectly on it's point in a room with no wind. While this article says that they found 13 solutions, none of these solutions would actually work because space isn't perfectly smooth. It's full of objects with gravity. So, like the balanced pencil that gets knocked over by a slight gust of wind, a passing rogue planet would throw any of these "solutions" out of whack and they would progressively destabilize. Where as a stable system (sun/earth/moon) can hande some perterbations and remain stable.
This diagram below is an example of a perfectly stable 3 body orbit in a computer program, but like the pencil, a small push on any one of the 3 objects would send it towards instability.
Hope that's not too long. I can clean up if needed.
The three body problem is a theoretical problem in Newtonian mechanics. It is possible to solve, exactly, the two body problem: Both bodies move in conic sections, typically ellipses, relative to their common centre of mass. The general three body problem cannot be solved exactly. There are special cases that can be solved exactly, of greater interest are situations in which, although the equations cannot be solved exactly, the system is stable in the long term.
There are several possible configurations:
The "Sun and planets" configuration: In this configuration one body is much more massive than the others, it is relatively fixed, and the second and third body orbit it. The second and third body are in orbits that remain sufficiently distant from each other that they remain in their orbits around the sun.
The sun-planet-moon configuration In this configuration, one body is much more massive than the other two. The second and third body are in a close orbit around each other.
The circumbinary configuration in this configuration, two large bodies are in a very close orbit around each other, the third body is distant, and orbits much further out. This is the "tattoine" configuration, with two suns close to each other orbited by a distant planet.
No real system is a theoretical three body system, since there are always perturbations (from non-spherical objects, from tidal distortions or from the fluctating gravitational field of the rest of the universe) But there are lots of examples of systems with more than two bodies.
The solar system provides examples of all these configurations. The system of "Sun-Earth-Moon" is a three body system. The system of Sun+planets has at least 9 bodies, in orbit and stable for billions of years (although not proved stable, and known to be chaotic over large timescales). Circumbinary systems are rarer (as they are less stable) but [Kepler 16b] (https://en.wikipedia.org/wiki/Kepler-16b) provides an example of one.
What is common to all three configuations is that they can be approximated by two body solutions. For example, the moon orbits the Earth in an elliptical orbit, that approximates a two body system, and the Earth-moon system orbits the sun in an elliptical orbit too.
What you won't find in nature is three bodies in a highly chaotic orbit which deviates greatly from an elliptical orbit, nor will you find orbits like the figure-of-8 orbit, as these orbits are not stable. In a fairly short amount of time either on body is ejected, or two bodies collide.
There are many systems with 3 and more bodies, but more or less universally they are highly hierarchical systems that behave approximately like a bunch of two-body systems. For example:
The solar system has 4 giant planets, 4 rocky planets, dozens of moons, and thousands of asteroids and Kuiper belt objects. Most bodies approximately behave like they are just orbiting the Sun, and the moons approximately behave like they are just orbiting their host planet. Celestial mechanics is largely the study of the higher order corrections on top of that.
There are triple-star systems, but they usually have one close binary, and a third, more distant star. The close binary behaves mostly like the two stars just orbit each other, and the distant third star approximately behaves like it just orbits the center of mass of the binary.
A galaxy has hundreds of billions of stars. But they are orbiting their common gravitational potential. In the very distant future most of the stars in the galaxy will be shed away, but that will take much longer than the present age of the universe.
These kinds of systems are often dynamically unstable, but if the stability timescale is longer than the age of the universe, it doesn't matter.
Yes, there are triple star systems, bound by gravity, that seem stable. See a list here:
There are also systems with 4 stars or more, although these are fewer, of course.
One could argue that star clusters and galaxies are the ultimate N-body problem, for very large values of N.