Is it possible to interpolate $e^n$ in more than one way? The most basic definition of exponentiation is repeated multiplication, $$e^n = e \cdot e \cdot  \cdot  \cdot  \cdot  e$$ $n$ times 
However, if $n$ is a rational number such as $2.4$, this definition breaks down. One way we've been able to interpolate the function $e^x$ to include all real values is by using infinite series $$e^x = \sum_{n = 0}^{\infty} \dfrac{x^n}{n!}$$ 
My question is, is it possible to find another function $f$ which is smooth over all the reals and agrees on integer values with $e^x$ but disagrees on rational values?
If it exists, could we use such a function to completely rewrite/overhaul a huge portion of all the math that exists today?

I believe there are tons of such functions. One of them is:
$$e^x \cos (2\pi x)$$
