How can a layperson learn to intuit exponential growth?

Question

I edited and shortened the following's origin. Is learning to intuit exponentiality, implausible (e.g. like hearing Xenakis's gratingly dissonant music as Mozart's tonal consonant melodies)?

Problem 1: Exponents aren’t intuitive

Math and statistics, aren't natural. Here’s an example: What’s the chance of getting 10 heads in a row when flipping coins? The untrained brain might think like this:

“Well, getting one head is a 50% chance. Getting two heads is twice as hard, so a 25% chance. Getting ten heads is probably 10 times harder… so about 50%/10 or a 5% chance.”

And there we sit, smug as a bug on a rug. No dice bub.

After pounding your head with statistics, you know not to divide, but use exponents. The chance of 10 heads is not $.5/10$ but .5¹⁰ $ \approx .001$.

But despite training, we still remain caught again. At 5% interest we’ll double our money in 14 years, rather than the “expected” 20. Did you naturally infer the Rule of 72 when learning about interest rates? Probably not. Understanding compound exponential growth with our linear brains is hard.

Answer 1

In the 1970s in the US, there was a commercial for Faberge Organics shampoo that illustrated the power of doubling. The story was that a woman liked her new shampoo so much that she told two friends about it, and those two friends each told two friends, and so on... With each additional "and they told two friends", there was a doubling of the number of identical women shown.

This was spoofed in the 1992 movie Wayne's World where the Wayne character is trying to explain how guys spread information.

Wayne is talking to a woman.

"Well, you know how these things start. One guy tells another guy some'n', an' he tells two friends..."

Two Waynes are shown

"and they tell two friends..."

Four Waynes

"and they tell their friends..."

Eight Waynes

"and so on"

Sixteen Waynes

"and so on"

Thirty-two Waynes

"and so on"

Back to Wayne and the woman

"You know how these things go."