This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.
The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.
A related, and more generic, question is:
How do I change this diagram/image/equation/formula so that it represents <something else>?
Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.
Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?
Thanks to Michael Pershan for the idea.