• The Vatican (a "country" on paper; but seriously?)

• The Sovereign Military Hospitaller Order of Saint John of Jerusalem, of Rhodes and of Malta (not to be confused with the country of Malta; this lot have no territory)

• The Organisation of African, Caribbean and Pacific States

• The African Development Bank

• The Andean Community (another free trade area)

• Both the Commonwealth of Independent States (former soviet bloc) and the Commonwealth of Nations (former British Empire)

• The International Olympic Committee

…

The list of organisations sending delegations is pretty big, even just limiting to the ones important enough to get into Wikipedia; you can probably find a whole bunch more of small ones.