If you were like me as a kid, you probably punched some numbers in a calculator and wondered, what happens if I just keep pressing the equals sign? Leonard Euler asked himself a similar question: What happens if I keep raising a number to a power over and over again? One might think the result would simply tend toward infinity. However, Euler showed that there are some numbers which, under the operation of infinitely repeated exponentiation, produce very interesting results. In this talk we'll explore these numbers, and then use the work of Lambert and Lagrange to extend our ideas about infinite exponentiation into the world of complex numbers. Interesting math history, fun with calculators, and beautiful pictures are guaranteed.