An interesting experiment is to ask people how many chords there are in music. You'll be surprised to find out that most musicians don't do any better at answering that question than non-musicians. Why do you suppose is that? It is probably because it sounds like one of those questions such as "How many grains of sand on the seashore are there?", or "How many stars are there in the sky?" And in a sense it is, but in another sense, we can get a fairly accurate sense of chord population just by calculating all the chord types and then multiplying them by the number of inversions that are possible and the number of octaves that are possible on any given instrument.
So let's start with a listing of chord types: Major Minor Diminished Augmented Diminished 7th Major 6th Minor 6th Major 7th Minor 7th Half-diminished 7th 9th Flat 9th Sharp 9th 11th Sharp 11th Suspension 13th Sus 7th Aug 7th 9th/Major 7th 6th/9th Add 2nd Add 4th Flat 5th 7th with flat 5th That's 25 of the most-used types. There are several other variations, but these chord types will do nicely for our purposes of estimating the total number of chords. Each chord can be inverted -- turned upside down -- by the number of notes in the chord. For example, a 3 note chord has 3 positions -- root position, first inversion, and second inversion.
A 4 note chord has 4 positions, a five note chord has 5 positions, and so on. We will say for arguments sake that 4 positions is the average, knowing that some chords have more and some have less. So if we multiply 25 chord types by 4 positions, that gives us 100 possible chords per octave. Some instruments only have the range to play 2 or 3 octaves, whereas a piano with its 88 keys can play 7 octaves -- 100 chords in the lowest octave, 100 chords in the next octave, 100 chords in the next octave, and so on up to the top octave of the keyboard. So on the piano we could theoretically play those 100 chords in all 7 octaves, giving us 700 possible chords. Of course, some would sound so low or so high that they wouldn't really be useable in a song.
But still, they are possible. So what's the answer to the original question? It depends upon the instrument and how many variations of each chord the individual musician uses.
A series or free lessons from Duane on the various aspects of piano chords & the blues is available: "Exciting Piano Chords & Sizzling Chord Progressions!"