As of writing this, 11 October 2023, I am reasonably certain that the star systems with the highest multiplicity, whose orbital periods of each of the stars are known, are those of the two septuple systems Nu Scorpii and AR Cassiopeiae. For a breakdown of their orbits, see this answer to the question How do the orbits of Nu Scorpii and AR Cassiopeiae work?
Further, according to Wikipedia, there are three new star systems of equal or higher multiplicity (and my Google searches do not yield me any results). They are a septuple named V871 Centauri, an octuple named Gamma Cassiopeiae, and a nonuple named QZ Carinae. And, it would appear that not all of the orbits are well understood an all three of these systems.
The only source I could find describing V0871 Centauri's orbits, Multiply eclipsing candidates from the TESS satellite, published 8 May 2022, says "V0871 Cen is probably a septuple-star system of architecture (Aa-Ab)-B-C-D." It continues:
The most inner two pairs are the 2.8 and 2.09 days binaries (we name it Aa-Ab), accompanied with a more distant component B with its only poorly-constrained orbit by Zasche et al. (2009). And much more distant C and D components are probably bound (due to their similar proper motion), but only very weakly.
So, while orbits of the inner binaries are understood, the more distant stars' orbits are not.
Then for Gamma Cassiopeiae, I couldn't find any source to describe its orbits, and the only source I could find confirming it's an octuple star system was Surveying the Bright Stars by Optical Interferometry III: A Magnitude-Limited Multiplicity Survey of Classical Be-Stars, published 15 September 2021. Now this one is really technically dense but, so far as I can tell, it does not give any orbital periods.
Finally, for Qz Carinae, the only source I could find describing its orbits was Towards a consistent model of the hot quadruple system HD 93206 = QZ Carinæ, published 15 April 2022, which explicitly states that it only has orbital information from the four inner, dual binary star systems.
Given that none of the other star systems known to have a multiplicity equal to or higher than Nu Scorpii and AR Cassiopeiae have all of their orbits known, as shown here, I am confident that these two systems currently have the highest known multiplicity, whose orbits are known. And, I can't thank HDE226868 enough for their answer.