I usually can only come up with two separate harmonies for a song, the ones I came up with after that are usually an octave higher or lower than those two harmonies.
4 voices are common. The classic "Glenn Miller" sound had the melody doubled on tenor sax and clarinet an octave apart, with the three other saxes harmonising within that octave. So that's 4 independent parts. Also look at the "locked hands" piano style associated with George Shearing.The Baroque composers regularly wrote 4-part counterpoint (that's even cleverer than just adding parallel harmony lines) and sometimes demonstrated they could manage more! Even a simple hymn tune is normally harmonised in 4 parts - and the aim is to give each voice a coherent melodic line to sing.
Taking that a nominal chord will have 3 notes - root, 3rd and 5th, before getting to the octave of the root, it's 3. Even with first or second inversion. However, 2nds (or 9ths), 6ths and some sort of 7ths will also fit with the original triad. Not all at the same time, but 1,2,3,5 works. As does 1,3,5,6, or 1,3,5,7. Going over the octave, but not duplicating notes, there's a 9th chord harmony, with 1,3,5,7 and 9. Of course, the 3s can be major or minor, the 5ths P, # or b, the 7ths major or minor, and the 9ths # or b. Then there's 11ths and 13ths. So, quite a few, and I expect I've missed some. Take a look at some guitar chords, with close voicing especially, for ideas.