At what point does adding another note to a chord become chaotic or
unpleasant to the ear?
Dissonance is the musical term, but adding more notes is not necessarily the factor which will produce dissonance. Add enough pitches and your get white noise which some people like to fall asleep to!
If we are talking about traditional, functional harmony, I would say 5 tones (ninth chords) is the limit. After that, we get 11th chords which begins to undermine tertian harmony and takes us into quartal harmony. Not necessarily unpleasant, but perhaps chaotic from a very traditional musical perspective.
What known chord has the most notes...
A lot of avant garde music will direct each player in an ensemble to play freely where you wouldn't be able to get an exact count of pitches on the resulting chords. In that sense we are facing an 'uncountable' idea.
Tone cluster is the name given to adjacent scale notes played together. Clusters can contain so many notes there is a special notation for them...
There is some similarity between tone clusters and quartal harmony as quartal chords can be inverted into clusters.
One could argue these avant garde devices are not proper chords but rather contrapuntal and percussive effects.
...and is there way to hear that chord here as an answer?
The question kind of supposes such music is unheard of, but there is a lot of music using these kinds of dense sonorities. That music doesn't get much air time on the radio or many recital hall performances, but it exists. Is it unpleasant? That's tricky. It's a matter of taste and especially exposure. If one has only heard modern and avant garde music in horror movie soundtracks, you have been conditioned to react a certain way.
An example of tone clusters is...
By way of comparing levels of dissonance you might trying listening to...
Debussy's music is normally regarded as very beautiful. What the West Windw Saw surely is dramatic and in places very dissonant.
I think Cowell's Manaunaun uses comparatively softer dissonances even though the tone clusters are numerically denser than Debussy's chords.