It's difficult to say. A particular set of notes can conceivably be called by many different names. By your rules, and following the generally accepted conventions of chord symbol rules, what would we call D♯ and F played together? If we respelled F as E♯, it could be called D♯(sus2omit5) or D♯(add9omit3,5). Do those count separately?
Then, are we talking about the note D♯ or the piano key sometimes called D♯? If it's the latter and we kept F as F, we could also include E♭(sus2omit5), E♭(add9omit3,5), and F7(omit3,5). And how about G♭♭7(omit3,5)?
I could go on, but what I can't do is think of a properly named chord that has just D♯ and F in it. Yet, how can anyone say those two notes can't be sounded together? You see the problem?
So if a chord is defined as 2 to 10 unique pitch classes and inversions don't count, there would still be additional things to take into consideration, and counting the combinations is way more than simple math. You need to consider the compatibility of each interval. You can't, for example, have a diminished 5th and an augmented fourth in the same chord, can you? That's the same note. But let's say we define a chord as a root plus one or more of the following intervals:
- A second, either minor or major.
- A third, either minor or major.
- A fourth, either perfect or augmented.
- A fifth, either diminished, perfect, or augmented.
- A sixth, either minor or major.
- A seventh, either diminished, minor, or major.
- A ninth, either minor, major, or augmented.
- An eleventh, either perfect or augmented.
- A thirteenth, either minor or major.
...you would come up with 46655 chords for each root note. But before you go multiplying that by every root, you'd have to eliminate chords which don't make sense. A minor third with an augmented fifth is really just a different major chord, so let's reject those. You wouldn't want something like a Csus2add9 chord, because that's duplicating a pitch class that's already in the chord. You wouldn't want something like Cadd♯9omit3 because that ♯9 is better identified as a minor third. So, if you take away everything that doesn't make sense, you're down to 7637 possibilities, by my count. Many of these are enharmonic, but should still count separately. For example, C+7add♯4 and C7♭5add♭6 are different by interval, but the same by piano keys depressed.
Whatever the count, you'd have to multiply that by each possible root, of which there are 12. If you count typical enharmonics, there are 17. If you count B♯, C♭, E♯, and F♭, then there are 21. If you count double sharps and flats, that's 35 different root notes. 35 times 7637 (if you trust me on that) is 267,295 different chords. Again, you'd have tons of enharmonic chords.
This is an interesting theoretical question. I wrote an algorithm to explore the possibilities and I asked a few questions here to help me do it. (See this, this, and this if you're interested.) You can check out the fruits of my labor here: