To give a rough estimate, there are approximately 12 (notes) * 3 (types of chords) * 4 (inversions) = 144 different piano chords. However, this number does not include the possibility of using extended chords, such as seventh chords, ninth chords, eleventh chords, and so on. When including these extended chords, the number of possible piano chords becomes even larger.
In conclusion, there are roughly 144 basic piano chords, but the actual number of possible chords is much larger when considering extended chords.