Re: Exponential growth ... a^t, for a an integer
any real number greater than 1
Why stop there? For reals in (0,1), you have exponential growth in the value's smallness. For 0 and 1, you have exponential growth in the value's sameness. "This 1 is much more the same than it was yesterday!"[1]
Daft explanations of exponential "growth" for negative reals[2] and complex numbers with non-zero imaginary parts are left as an exercise for the reader. Many of whom, I'm sure, could use the exercise.
[1] Silly as that is, it reminds me a bit of Matt Skala's famous essay "What Colour are your bits?", where he considered how people may assign different interpretations to identical values based on their provenance.
[2] Pretty easy for integer exponents.