Wikipedia's article on the Mother Goose rhyme "As I was going to St. Ives"Wikipedia's article on the Mother Goose rhyme "As I was going to St. Ives" suggests that curiosity about higher powers has existed a long time.
The Rhind Mathematical Papyrus (Problem 79), dated to around 1650 BC, could be described with the following text:
There are seven houses; In each house there are seven cats; Each cat catches seven mice; Each mouse would have eaten seven ears of corn; If sown, each ear of corn would have produced seven hekat of grain. How many things are mentioned altogether?
Edit: Interestingly, the $7^5$ hekat of grain here are counterfactual (imagined, not actually existing); as is perhaps the very notion of a number like $7^5$, to the author of the papyrus.