What is the upper bound of number of stable interacting stars in a star system?
There is no true upper bound. You could argue that a galaxy is a vast multiple star system, with hundreds of billions of stars. Natural galaxies are not perfectly stable (random encounters eject stars over very long timescales), but one can construct stable systems by taking pairs of similar-mass binaries and placing them into a remote circular orbit around each other. This can then be repeated indefinitely, with $2^n$ stars forming a vast system.
In practice such neat systems are unlikely to form naturally (4 levels seem to be the max observed and often involves very unequal masses) and very wide systems will encounter other stars often enough to be destabilized. Eccentricities tend to have a triangular $f(e)=2e$ isothermal distribution, making things less stable. See (Duchêne &amp; Kraus 2013) for more statistics.
(As an aside, Jaron Lanier and Piet Hut have suggested constructing such vast hierarchical systems as a clear indicator that there is intelligent life present.)