The seventh is not the only note you can add to a basic triad to get a new chord. You can continue to extend the chord by adding to the stack of thirds (Section 5.1), or you can add any note you want. The most common additions and extensions add notes that are in the scale named by the chord.
The first, third, and fifth (1, 3, and 5) notes of the scale are part of the basic triad. So are any other notes in other octaves that have the same name as 1, 3, or 5. In a C major chord, for example, that would be any C naturals, E naturals, and G naturals. If you want to add a note with a different name, just list its number (its scale degree) after the name of the chord.
Many of the higher added notes are considered extensions of the "stack of thirds" begun in the triad. In other words, a C13 can include (it's sometimes the performer's decision which notes will actually be played) the seventh, ninth, and eleventh as well as the thirteenth. Such a chord can be dominant, major, or minor; the performer must take care to play the correct third and seventh. If a chord symbol says to "add13", on the other hand, this usually means that only the thirteenth is added.
You may have noticed that, once you pass the octave (8), you are repeating the scale. In other words, C2 and C9 both add a D, and C4 and C11 both add an F. It may seem that C4 and C11 should therefore be the same chords, but in practice these chords usually do sound different; for example, performers given a C4 chord will put the added note near the bass note and often use it
as a temporary replacement for the third (the "3") of the chord. On the other hand, they will put the added note of a C11 at the top of the chord, far away from the bass note and piled up on top of all the other notes of the chord (including the third), which may include the 7 and 9 as well as the 11. The result is that the C11 - an extension - has a more diffuse, jazzy, or impressionistic sound. The C4, on the other hand, has a more intense, needs-to-be-resolved, classic suspension sound. In fact, 2, 4, and 9 chords are often labelled suspended (sus), and follow the same rules for resolution (pg 185) in popular music as they do in classical.