One of the challenging things about music theory is the fact that it is a moving target. In other words, it is not a clear cut discipline in the way that mathematics might be. In math, 2+2 will always be 4 regardless of the time period or culture. That is not true with music.
I know that is a bit hard to believe but it is true. What is right and wrong about music is based to a large degree on the perspective of a time period and culture. Certain things that are “right” (sounding good to us) today would have been “wrong” (sounding awful) not very long ago. Things that we consider “wrong” today will be considered acceptable at some point in the future.
The simple triad is an example of that. Few if any people alive today would consider the sound of a triad to be “wrong” but if you go back maybe eight centuries, that would not be the case. A triad would have sounded very wrong.
The reason I wrote those first three paragraphs is to give you a little insight on why I can’t answer a question I often get: how many chords are there? From a musical standpoint, I haven’t a clue. I can give you a close estimate on the number of chords that are used today but I have no idea how many legitimate chords I would be leaving out.
The funny thing is that from a math perspective, the number of chords is a simple logic problem (at least for those who like math). After all, a chord is a group of at least three notes. A chord can technically have up to 12 notes because there are 12 unique notes in a scale. The potential number of possible chords with between three and 12 notes is just a math formula. It is an enormous number by the way.
From a musical standpoint, the answer to how many chords there are is this: it is the small subset of the mathematical possibilities that sound good to us right now. In other words, my answer might be different from yours. My answer would definitely be different from someone that lived a century ago or someone who lives a century in the future.
Even musicians living at the same time but in different genres will disagree on the list. What sounds like noise to me might be an acceptable chord to a modern classical musician using a system like serialism. To make things more complex, chords are named differently by musicians in different genres. The rules change a bit across genres. For example, I might call a F triad with an added G by the name F(add 2). Somebody else might call it F(add 9) or Fsus2.
All in all, it is just a bit of a mess. We don’t have the equivalent of the periodic chart where generally all scientists agree on foundational principles of matter. Musicians hear different things, accept different things and then name those things differently. The differences are not small; they are in fact huge. One chord chart might list 350 different chords and another might list 150.
For all these reasons, I will readily admit I have no idea how many chords there are. However, I do know this: chord charts of the future will be much more extensive than the ones we have today and today’s charts are much more complex than the palate used by counterpoint composers a few centuries ago. If you go back much further than that, you would have just gotten a blank stare if you asked a musician for a definitive chord chart because chords were not even discovered.
Music theory is a discovery process. In that regard, it is sort of like science after all.