For me the only way to make sense of your question is to interpret it as "which chords outside the key are frequently added to a piece in major?". Because otherwise the obvious answer is that if you allow any note, any chord could be added.
In a major key, it is quite common to add chords from the parallel minor key. The concept of using chords from a parallel tonality is called modal interchange. The parallel minor key will give you chords with roots on all chromatic notes except for the b2 (the Db in the key of C) and the #4/b5 (F#/Gb). A chord with the b2 as a root can be borrowed from phrygian. If you combine the chords from the parallel (natural) minor scale and from phrygian, you get these additional triads in the key of C:
Db major | Eb major | Ab major | Bb major
Apart from these chords with root notes which are not part of the C major scale, you can also use chords with roots from the scale, but with other chord tones outside the scale. These other chord tones can also be borrowed from the parallel minor key, and the most frequently used chords are (again in C):
D diminished | F minor | G minor
Yet another source of chords from outside the major scale are secondary dominants (and their related II chords), resolving to diatonic chords:
A7 => Dm | [F#m7(b5) B7] => Em | [Gm7 C7] => F | D7 => G | E7 => Am
where the arrow => means 'resolves to'.
See also this answer to a related question.