Explaining to kids that $2^n$ can quickly yield high values I have to explain to kids that $2^n$ can yield high values. Not with math and numbers but with a story rather.
There was a story about a pharaoh and wheat.. 
 A: While asking my question, i remembered.

The Farao's architect.
When asked what he wanted paid, the architect said: "A chess board is 8 by 8 squares. Give me one grain (seed) on the first square, and for each square double the amount. So two on the second square, and four on the third square. And so on for the remaining squares."
"Very well", said the Farao, and his country soon went bankrupt.
Why?


A: A nice way for him to realise by himself is to let him start with one and multiply it by two.
Ok, that was easy. Multiply again by two.
Ok, still too easy, so multiply by two again and again.
When he finds it difficult because numbers are too large, ask him how many multiplications he did and let him compare with $2n$ so that he can realise how big he got in comparison.
Another similar way with money. Whatever your currency, start with one (hopefully it shouldn't be too much). Does he have this amount of money? Probably yes.
Multiply this amount by two over and over again. Pretty soon he certainly won't have enough money!
A: An interesting interactive one could be breaking a rock or stick in half a bunch of times and showing how quickly there are a lot of pieces. I personally would find it awesome if a someone lectured me on how $2^n$ was big by hitting a rock repeatedly.
