I suppose you're asking about what the largest *named* chord is. You can have a chord with as many pitches as you like, across the whole range of the instruments playing that chord, and of course even more if you are using quarter-tones and micro-tones. However, as you say, the largest number of different pitches you could have with the "normal" chromatic system is 12.

You are right that chords being described with common naming conventions (built in thirds for instance) usually have no more than 7 distinct pitches, and so the 13th chord is the limit. Interestingly, one feature of this way of describing chords is that these chords have no semi-tone clusters (i.e. three adjacent semitones).

Beyond this, and for chords with semitone clusters, I would use PC Set analysis to describe sets of pitches.

There is one exception. Chords based upon a full Octatonic (diminished) scale, have no semi-tone clusters, but do have *eight* distinct pitches. These can be described using conventional naming, too, and are used in, for instance, Jazz.