I am an avid listener of the RoosterTeeth Podcast.
In a not-so-recent episode of said podcast, somebody asked the question of why there are so many trilogies in our modern media (books, movies, video games). He sited examples like the Hunger Games, The Matrix, etc., and how they all seemed to be complete stories within their first part. So why make more, and, more importantly, why specifically 3 parts?
A valid question, which I, too, have often wondered.
However, the answer is actually pretty logical, and it dawned on me while reading Essentials of Screenwriting by Richard Walters. Some literature major, somewhere in the world, could probably refute me on this is he/she saw fit, but it seems that the abundance of story trilogies can be accredited to (or blamed on) Aristotle. It was actually Aristotle that defined the formula of the modern day narrative:
A complete story has a beginning, a middle, and an end. Three parts.
That formula typically should apply to a whole movie or book, otherwise you leave the audience unsatisfied. However, as anybody knows, there are many stories whose whole stories are too large to fit into one movie or book. So, considering that an entire story arch has to have a beginning, middle, and end, it's often more natural for it to fit into 3 segments, as opposed to 2 or 4.
Is this a rule? Of course not. There are many franchises that are well done, which have 1, 2, 4, or 8 parts to them. In most cases, it seems, these numbers work best for the kind of content where each part is a new story in the same universe, rather than one carrying directly into the next, but it can work either way. The trilogy is merely the format that is tailored specifically for how stories are written and told.
I think we can all agree that there are these exceptions on both sides. There are trilogies that should have only been left at one movie, and there are movies that should have been continued but fell short of their potential (Buckaroo Banzai, anyone?). However, no matter how you look at it, "to confuse the exception for the rule is to fall on your face." - Richard Walter (Essentials of Screenwriting).
That seems to be the logical explanation for the surplus of trilogies.