Is there any combination of up to six notes in the chromatic scale that could not be classified and named as a chord? Can anyone give me an example and explanation if so?
No, and for at least three reasons:
Assuming "chord" to be a tonal entity, we can explain anything as having alterations, omissions, and extensions. With add11, ♭13, no5, etc., we can make sense of any combination of tones.
We can understand harmonies as combinations of chords; such polychords allow any and all possibilities.
We have systems of understanding "chord" that do not assume tonality. Pitch-class set analysis—a system in which we assign integers to members of a harmony—is perhaps the most common. No matter how wild of a chord you come up with, there is a pitch-class set label for it.
Depends on your definitions. There are certainly pitch sets that would be difficult (and pointless) to label in the 'C, Gm7, F#m7(b5)(b9)' naming system, or that defy functional analysis in the 'bii7 of iii' way. But some will say that ANY pitch set is, by definition, a chord. And, as @Richard says, any pitch-class set can be labelled.
Is there any combination of up to six chromatic notes that could not be classified and named as a chord?
From the point of view of naming and classification, some would consider that groups/sets of 2 notes aren't named 'chords' as such: A chord is three notes? What do you call just two notes?.
Interesting question. I would submit that if we take chord theory and apply it to pitches either above or below the ranges of human hearing that the resultant Chords would no longer exist simply because we can't hear them and therefore they would never be played. My thinking is that music isn't really music until it's being played, but that's just my own thoughts on the matter.
A chord can be defined as several notes sounding simoultaneously. No matter what notes you use. Some refer to clusters though, if you arrange the notes in close position, neighboring each other within the same octave.
For example a Cmaj7 chord uses a half step from 7th to root. You can raise the 7th an octave up and the chord loses its "cluster" character.
You could change octaves of your 6 note chromatic row, too and see what results you get.
A good orientation is the Schoenberg, Berg, Webern connection called "Zweite Wiener Schule" around the beginning of the 20th century. They do a lot with more or less mathematical constructions over tone rows. Even Bernstein´s West Side Story uses tone rows. It is impressing what he gets out of symmetric scales in terms of arranging, themes and chords...