The story
Blend of Brazil + -illion. Possibly originating from an old joke mocking former US president George W. Bush: Donald Rumsfeld is giving the President his daily briefing on Iraq. He concludes by saying: "Yesterday, seven Brazilian soldiers were killed in an ambush." "Oh, no! That's terrible!" the President exclaims. His staff sit stunned at this display of emotion, nervously watching as the President sits, head in hands. Finally, the President looks up and asks, "Um... how many is a brazillion, exactly?"
Etymology from Wiktionary, CC BY-SA 4.0.