As in, has there ever been an instance of three or more factions fighting each other on multiple fronts without any allies?
Depends what you mean by faction? Some religious and civil wars had more factions, sects, etc than you've had hot dinners. The problem is with your "multi-participant" war notion is that however temporarily parties will usually unite into two groups even if just temporarily to eliminate the other major group, then go and return to a two way war.
The Musket Wars in New Zealand are an example of chaotic intertribal warfare with constantly shifting alliances etc which might fit your definition? Your comprehensive source there is R.D. Crosby's The Musket Wars: A History of Inter-Iwi Conflict 1806-1845 (Oratia Media, 2012) sadly not online but it's on Amazon so it may be on Kindle.
hi! fyi, you'll find some examples in the FAQ (link on sidebar)