We usually come across close binary systems in Astrophysics. Also, we can have hierarchical systems in which there is a close binary. The Alpha Centauri is an example of an hierarchical triple.
But why don't we have any trinary or multiple systems?
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1https://en.wikipedia.org/wiki/HD_181068 is a system in which the orbital periods are not all that different (1 day for the two closest stars vs. 25 days for the third one). – probably_someone Nov 27 '19 at 13:08
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1Who says we don’t? – Kyle Kanos Nov 27 '19 at 13:15
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1Related: https://physics.stackexchange.com/q/321760/ – Kyle Oman Nov 27 '19 at 14:16
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Would you consider something like a globular cluster ($n$-body) to be a 'system'? – Jon Custer Nov 27 '19 at 18:08
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Non-hierarchical multiple star systems (where all members have similar masses and similar orbital periods) are possible in theory, and occasionally observed in reality. Their relative rarity compared to hierarchical systems suggests that non-hierarchical systems may be unstable (or, at least, have very small domains of stability) and so mostly short-lived. However, this seems to be an ongoing area of research.
gandalf61
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1Might be worth mentioning that there's an easy analytical solution for two bodies, but there's no analytical solution for more than two, and 3+ body systems often are chaotic. – Solomon Slow Nov 27 '19 at 16:14