Theoretically, it is possible to have a major 27th chord, a minor 35th chord, etc.
No it isn't, not by the principle of stacking thirds, because the next third above the thirteenth is the same pitch as the root, and every third above that will repeat some pitch you've already encountered. (Keep in mind that chord theory generally ignores the voicing of a chord except for the identity of the bass note, so C3 B3 G4 E5 is just as much a Cmaj7 chord as is C4 E4 G4 B4.)
In any event, there appears to be a misconception underlying this question:
Consider chord qualities: major, minor, diminished, augmented, dominant, half-diminished, etc. To build a chord of quality Q, stack an appropriate number of thirds from the scale of quality Q.
This works for major and minor chords, more or less, but that is just a coincidence. There aren't diminished, augmented, dominant, or half-diminished scales.
Historically, chords were not thought of as stacks of thirds but as a collection of intervals relative to the root. A major triad comprises a major third and a perfect fifth. A minor triad comprises a minor third and a perfect fifth. A diminished triad has a diminished fifth and a minor third. An augmented triad has an augmented fifth and a major third. In each of these cases, the triad is named after the one interval that is most significant in establishing the chord's character and identity.
Similarly, seventh chords are primarily named after the seventh, with some implicit secondary attention being paid to the third and the fifth. The "big five" are the major seventh chord (a major triad plus a major seventh), the minor seventh chord (a minor triad plus a minor seventh), the dominant seventh chord (a major triad plus a minor seventh) the diminished seventh chord (a diminished triad plus a diminished seventh), and the half-diminished seventh chord (a diminished triad plus a minor seventh, called "half diminished" because a diminished seventh chord has two diminished intervals while the half-diminished seventh chord has only one).
All of these chords can be derived from the same scale, the diatonic scale, except for the diminished seventh and the augmented triad. Consider the diatonic scale that comprises only the white notes on a keyboard. If you build a triad on any of the seven pitches, you get a major, minor, or diminished triad. The major triad is found on C, F, and G, the minor triad is found on D, E, and A, and the diminished triad is found on B. As for seventh chords, the major seventh chord is found on C and F, the dominant seventh chord is found on G, the minor seventh chord is found on D, E, and A, and the half-diminished seventh chord is found on B,
Now this diatonic scale with only white notes is commonly known as the C major scale, but it can also be the A minor scale. In that case, however, the sixth and seventh scale degrees will sometimes be raised chromatically. In other words, you'll sometimes have F♯ instead of F♮ and sometimes G♯ instead of G♮. This gives you some more possibilities, such as a dominant seventh chord on E, and, appearing for the first time, a diminished seventh chord on G sharp and an augmented triad on C.
This description reflects the historical development of harmony much more accurately than does the idea of building chords by stacking thirds on one another. One aspect of this that I glossed over at the beginning is that triads first arose more as a third added to a fifth than as a fifth added to a third. That is, people first regarded G and D (for example) as a complete and stable combination, and only later did they begin to include B♮ or B♭ in between.
The idea of building increasing stacks of thirds arose in the late 19th and early 20th centuries when music was becoming increasingly chromatic and acoustically dissonant chords such as seventh chords began to be treated as harmonically stable consonances. Theorists needed a framework to explain why sevenths and ninths and fourths and sixths were not being treated as dissonances that required resolution but as stable sonorities. Third stacking explains this fairly well, so all of a sudden instead of fourths and sixths we started having elevenths and thirteenths.
Now, back to "to build a chord of quality Q, stack an appropriate number of thirds from the scale of quality Q." This seems to work, but as I said, it's just a coincidence. It may seem as though major and minor chords are so named because their pitches come from the major and minor scales, but it's more accurate to say that the scales (and the keys) got their names from the same source as major and minor triads: they're all named after the quality of the third.
