Title pretty much says it all, but I have a couple of additional questions. 1) Is it even possible?
2) If it is, then where do we start? Like what is the definition of $e$? Is it possible to do this without defining $e$ as the solution to $f'(x) = f(x)$, or as the limit $n\to\infty$ of $(1+1/n)^n$?
3) If $x,y$ were real, is it possible to prove the claim without using complex numbers or real analysis?
Thanks!
For complex numbers I would start with $\displaystyle \exp(z)=\sum_{n=0}^\infty \frac{1}{n!}z^n$.
Then use the Cauchy product to have $$\displaystyle \exp(z)\exp(w)=\left(\sum_{ n=0}^\infty \frac{1}{n!}z^n\right)\left( \sum_{k=0}^\infty \frac{1}{n!}w^k \right)=\sum_{n=0}^\infty \frac{1}{n!} \sum_{p=0}^n \frac{n!}{p!(n-p)!}z^pw^{n-p}=\sum_{n=0}^\infty \frac{1}{n!}(z+w)^n=\exp(z+w).$$