It is perfectly possible for a planet to have a stable orbit around a single star. There are many examples in our own solar system and in other star systems.
It is perfectly possible for a planet to have a stable orbit around one star in a binary star system. There are many examples in other star systems. That is called a non-circumbinary or S-Type orbit.
It is perfectly possible for a planet to have a stable orbit around both stars in a binary star system. There are many examples in other star systems. That is called a circumbinary or P-Type orbit.
And beyond that it is hard to say.
Kepler 64 is a quadruple star system, with four stars, but the planet Kepler 64 b, or PH1b, orbits only two of those stars.
The two stars orbited by the planet, Kepler 64 Aa & Ab, should be very close to each other, only a few million kilometers apart, and have an orbital period around each other of 20 Earth days. The planet Kepler-64 b orbits them at a distance of about 0.34 AU with a period of 135.506 Earth days. The other two stars in the system, Ba & Bb, orbit each other at a distance of 60 AU and are separated by about 1,000 AU from stars Aa & Ab that the planet orbits.
Stars in multiple star systems are arranged in a hierarchal manner.
In a physical triple star system, each star orbits the center of mass of the system. Usually, two of the stars form a close binary system, and the third orbits this pair at a distance much larger than that of the binary orbit. This arrangement is called hierarchical. The reason for this arrangement is that if the inner and outer orbits are comparable in size, the system may become dynamically unstable, leading to a star being ejected from the system.
If you desire for the planet in your system to be habitable, with an oxygen rich atmosphere breathable by humans, your are out of luck.
The Earth is 4.6 billion years old, but has only had a breathable oxygen rich atmosphere for about 0.6 billion years. The oxygen in the atmosphere was produced by photosynthetic organisms billions of years after life first appeared on Earth.
So to have a breathable atmosphere, your planet would have to have temperatures suitable for life for billions of years before it produced a breathable atmosphere. So it needs to have an orbit which changes slightly over billions of years while the star or stars change their luminosity very slightly over billions of years.
So there is a limitation on the spectral types, masses, and luminosities of stars in a star system which can have a habitable planet. Limits on the masses of the stars thus produces limits on their possible stable orbits around each other. Limits on their individual and combined luminosity puts limits on how close or far the planet can orbit to have habitable temperatures.
I also note that planetary orbits, like those of moons, can and do change over time, due to tidal interactions between moons and planets, and planets and stars, and between stars in multiple system.
Most moons of planets in our solar system experience tidal acceleration which moves them slowly farther from their planets.
Most natural satellites of the planets undergo tidal acceleration to some degree (usually small), except for the two classes of tidally decelerated bodies. In most cases, however, the effect is small enough that even after billions of years most satellites will not actually be lost. The effect is probably most pronounced for Mars's second moon Deimos, which may become an Earth-crossing asteroid after it leaks out of Mars's grip. The effect also arises between different components in a binary star.
There are two classes of moons of planets, or planets of stars, which can experience tidal deceleration, causing them to move closer and closer to their primary bodies and eventually be destroyed.
Fast satellites: Some inner moons of the giant planets and Phobos orbit within the synchronous orbit radius so that their orbital period is shorter than their planet's rotation. In other words, they orbit their planet faster than the planet rotates. In this case the tidal bulges raised by the moon on their planet lag behind the moon, and act to decelerate it in its orbit. The net effect is a decay of that moon's orbit as it gradually spirals towards the planet. The planet's rotation also speeds up slightly in the process. In the distant future these moons will strike the planet or cross within their Roche limit and be tidally disrupted into fragments. However, all such moons in the Solar System are very small bodies and the tidal bulges raised by them on the planet are also small, so the effect is usually weak and the orbit decays slowly.
Retrograde satellites: All retrograde satellites experience tidal deceleration to some degree because their orbital motion and their planet's rotation are in opposite directions, causing restoring forces from their tidal bulges. A difference to the previous "fast satellite" case here is that the planet's rotation is also slowed down rather than sped up (angular momentum is still conserved because in such a case the values for the planet's rotation and the moon's revolution have opposite signs). The only satellite in the Solar System for which this effect is non-negligible is Neptune's moon Triton. All the other retrograde satellites are on distant orbits and tidal forces between them and the planet are negligible.
Imagine a planet having a figure 8 orbit around two stars in a binary system, a more simple arrangement than having a figure 8 orbit around three stars.
So when a planet one star swings in to the zone between the two stars where their gravity is equal and switches between Star 1 and Star 2, it will be orbiting Star 2 in the opposite direction to its orbit around Star 1. It should be simple to draw a diagram showing the orbit around the two stars, showing that from the point of view of the diagram the planet orbits one star clockwise and the other star counterclockwise.
The two stars will probably rotate in the same direction. So the planet will orbit one star in the prograde direction, the same direction as the star rotates and be slowly pushed farther from that star. And the planet will orbit the other star in the retrograde direction, opposite to the star's rotation and will be slowly pulled closer to that star. Thus the planet's orbits around the stars will be slowly perturbed and move from the distances necessary for the switch off between stars to be possible.
If the two stars rotate in opposite directions, the planet will either orbit both stars in their prograde directions and be pushed outward from each of the stars, or orbit both stars in their retrograde directions, and thus be pulled closer to each star. And either process should eventually move the planet's orbits away from the distances needed for the switch between stars to happen.
Because the planet's mass will be a tiny faction of the two stars, it will have a tiny tidal effect on the stars, and so will be pushed out of its proper orbits very slowly.
But because the two stars in a system with a habitable planet will have a comparatively narrow range in mass compared to the mass range of all stars, the two stars will have similar masses to each other and thus will strong tidal effects on each other and push themselves apart comparatively fast.
Thus they probably would move out beyond the distance where the planet can switch between stars, leaving it orbiting only one star, long before the planet became habitable with an oxygen rich atmosphere.
So a science fiction story where Earth explorers visit a young binary star system where there is a planet in a figure 8 orbit around two stars would be possible, I guess. Such an orbit could not last long, and the explorers would be amazed that they found an example of one.
But a science fiction story where there was a habitable planet with an oxygen rich atmosphere in such a figure 8 orbit around two stars would be very almost totally impossible since such an orbit probably would not last long enough for the planet to become habitable.
Of course in the case of planet in a figure 8 orbit around two stars it would be easy to calculate the orbits and their future changes. The orbits of the two stars could be calculated as a simple two body problem, since the mass of the planet would be negligible. And the orbit of the planet around one of the stars could be treated as a simple two body problem, with the gravitational effect of the other other star then added as perturbation. And when the planet switched to orbiting the other star, the same would be true.
Of course, if there were three stars orbiting their common center of gravity at about the same distance, that would be a three body problem, which are very complex and hard to calculate.
But since all the multiple star systems which have any great age have the stars orbiting in a hierarchal manner, instead of orbiting the common center of gravity at similar distances, it seems obvious that only hierarchal triple or other multiple star systems can have long term stable orbits. In a hierarchal triple star system the orbits of two stars can be considered a two body problem, while the third star would be far enough from the other two that the common orbit could be considered a two body problem instead of a three body problem.
And possibly someone who knows a lot more about orbital mechanics and has computer programs to calculate orbits could try designing a long term stable figure 8 orbit around two stars, one which could last long enough for the planet to become habitable.
But I doubt anyone would try to design a long term stable figure 8 orbit around three stars.