Q:

Four-chord songs with more than four chords

The Axis of Awesome has a song called "4 Chords", a medley of various songs all written, or so it's claimed, using the same four chords.
Now, from what I understand, a chord is just a bunch of tones played at the same time so they sound like one. For example, a 300 Hz note and a 500 Hz note played simultaneously will be a 100 Hz sound (because 100 is the greatest common factor of 300 and 500) with a lot of nuance in it, and that's called a chord. So the songs in the medley all have the same chords, that is, the same sounds with the same… what I called nuances.
They don't.
Never mind the lyrics: there are clearly more than four sounds in any of the songs when sung: just listen to 'em. But even watching the keyboardist's hands on the electronic keyboard (for the one segment where you get to do so for any significant length of time (warning, foul language in that clip)), you can see he hits more than four chords.
What gives? Am I hearing and seeing wrong, or is my definition of chord wrong, or is Axis of Awesome lying, or is there some explanation of how the songs in this medley are considered to have but four chords even though they have more?

A:

tl;dr - your definition of chord is wrong.
Your initial assumption

a 300 Hz note and a 500 Hz note played simultaneously will be a 100
  Hz sound

is unfortunately incorrect.
When you play a 300Hz note and a 500Hz note what you will get is a 300Hz note, and a 500Hz note, and a 200Hz note (the difference between them) and an 800Hz note (the sum of the two frequencies) (Have a listen to the demos on this page for examples)
In addition to that, a chord can have any number of notes in it (typically on a piano it usually has 10 or less notes, and on a guitar you're usually expecting 6 or less)
And finally, a chord can be played in many different ways, so if I wanted to I could play a 4 chord song using 20 different variations of that chord.
Update After watching the actual video, the keyboard player is only playing 4 chords, and not really playing any variations of them. He is sometimes accenting certain notes more than the others, but the 4 chord shapes he is using are consistent throughout that segment. He is hitting and releasing keys, sure, but only as parts of the same chords.

A:

The chords are I, IV, V, and vi (not in that order) which together contain every note in the key.  So any extra notes could be considered to be part of one of those chords.  Regardless though the idea is not that the keyboardist plays only a specific set of notes.  It's that four chords (in the broader sense Dr Mayhem mentions, which includes inversions and so on) are the basis for the songs, forming the major progression and sound.  Other chords (if any) serve only to transition to one of the 4 main chords.  And of course this is only true for the portion of the song being played — they never get to the part of Don't Start Believing that uses a iii chord!
None of those song snippets could be played recognizably without one of those 4 chords.  But they could all be played without any other chords and be recognizable.

Summarizing some discussion from comments:
There are 7 notes in every Major/minor key.  Capital-numeral (e.g., IV) denotes the Major chord starting on the numbered note, and lowercase-numeral (e.g., iii) denotes the minor chord starting on the numbered note.  Every note in the key is used by at least one of the four chords above (I, IV, V, vi), meaning that any note you play in the key could be considered to be functioning as part as one of those 4 chords rather than independently or as part of a different chord.
In the key of C Major, any combination of any C's, E's, and G's could be considered a I chord.  (Similarly any such combination would be a V chord in F Major, and so on.)  Most commonly you'll have one of each note very close together, but it's not necessary.  You might consider individual notes part of an arpeggiated chord, but it's important to note that notes can be left out without making it "not a chord".  This general sense of "chord" we're using is limited to specific notes, but not specific combinations or positions of said notes.
So, one "chord" in the general sense covers an entire subset of chords in the specific sense.  The class "G Major chord" covers the specific "G4 B5 D5" chord as well as many many others — "G4 B6 D7", "G4 B5", "D3 G3 B4", etc.

