I have launched a massive chemical weapon attack in my house, ongoing as I write.
Fleas are a good example of an exponential process. One day, you think you might have gotten a bite but are not sure. The next day, you spot a flea. The next day, 5, the next day, your son complains that he can't work on the computer because fleas keep jumping on his ankles. Time to declare war! And that's why my house is currently being filled with toxic gas.
Where do those fleas come from? From a litter of kittens who were in my house for a few weeks. I knew they had fleas, but the fleas politely stayed on them and did not bother anyone else. I half-heartedly tried to bathe them and comb them with a flea comb, but my efforts were ineffective, and because of the youth of the kittens, there was no other possibility. After the kittens were gone, the flea eggs were left behind, and must have hatched in the last few days.