Algebra: A sequence from long ago...

I keep forgetting to post our very first challenge, from way back when. Here it goes:

The Challenge: Given the first few numbers of a sequence,

1, 2, 9, 64, 625, ...

can you deduce the next number? As always, there is a pattern if you look closely enough.

Hint: Sometimes when numbers increase extremely rapidly, we say they are growing exponentially...