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 (as you say in your question). However, as you also 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 usually have no semitone 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 semitone clusters, but do have eight distinct pitches. These can be described using conventional naming, too, and are used in, for instance, Jazz.
To give an example, two Octatonic scales (and so also Octatonic chords) have roots on C: one with a semitone as the lowest interval; one with a whole-tone as the lowest interval:
- C Db D# E F# G A Bb can be described as C13#9b9 - yes it's a bit of a mouthful, but it is completely legitimate, and certainly used. (You always have to decide which letter name is used twice in an octatonic chord; here it seems sensible to use D twice, as we describe it with #9 and b9.)
- C D Eb F Gb Ab A B I'm not sure how you'd describe this… (without forcing it to fit a description; any ideas anyone…?)
So, the first of these two Octatonic chords would be my candidate for largest chord, using the conventional system of building and describing chords.