Demonstrate the power of exponential growth.
From Wikipedia: "In the DES block cipher, 56-bit key is used, resulting in a relatively small key space of size 2^56, which was demonstrated in 1998 could be searched exhaustively in 56 hours with a desktop computer."
A 56 bit key may look like this: "A56!/g="
What if we added one character? "A56!/g=y"
It would take one and half year to break.
What about two more characters? "A56!/g=yP"
It would take 419 years to break.
A key with yet one more character ("A56!/g=yP2&") would result in 100000 year search.
From Wikipedia: "The story is first known to have been recorded in 1256 by Ibn Khallikan. Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed."
The weight of grains of wheat on the first half of the chessboard is about 279 tonnes of wheat. On the second half of the chessboard the numbers become just ridiculous.
Bacteria splitting in exponential progression. This could be a basis for a nice doomsday story. Alternatively, self-replicating nanomachines, computer viruses, memes, Malthusianism etc.
Let's say evil nanomachines are replicating swiftly. People try to stop them, but they can only destroy them with linear efficiency: X tonnes of nanomachines per hour. The machines themselves, however, double their mass each hour. The deluge cannot be stopped. Exponential growth always trums the polynomial (in our case linear) growth.
Note: Discussion of herd immunity, vaccination and modelling of infectious diseases can be added here. It's kind of tangential, but not totally irrelevant (think computer viruses). Decide whether to include it or not.