I assume you mean more than one "mode" (scale pattern, relative to the tonic) rather than one modulation (which may simply be a key change). And I don't know if you'd count tunes which shift dynamically between a major mode and its relative minor mode (such as between C major and A minor, or C Lydian and A Dorian)--though that is similar to what happens in the example you cited.

I can't think of any 3+ mode examples offhand, though contemporary pop pieces (particularly in the jazz world) may flit through a variety of tonalities and "modes", even if standard notation wouldn't overtly indicate this. And of course, pieces in "minor" tend to use three scale patterns simultaneously: natural minor (Aeolian), harmonic minor (Aeolian with major 7th) and melodic minor (major 6th and 7th ascending, natural minor descending).

Now that I think of it, blues tunes, with fluctuating/quarter-tone 3rd and 7th degrees--and gapped scales (gaps sometimes filled in different ways by passing notes or ornamentation)--, might fit your description.

I wrote a tune to "Unhappy Bella" which constantly shifts tonal base and must use at least three "modes" (according to old-style theory); but as it doesn't settle in any key or mode long enough for one to be firmly established, so I don't think it's the kind of tune you have in mind.