I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges towards the $\exp$.
Let : $$f\left(x,h,c\right)=\left(1+\frac{x}{c^h}\right)^{c^h}\text{(A quite understandable approximation for the exponential function)}$$ $$q\left(x,h,c\right)=\sum \limits_{p=0}^h\frac{c^{\frac{p^2+p}{2}}}{\left(\prod\limits _{i=1}^p\left(c^i-1\right)\right)\prod\limits _{i=1}^{h-p}\left(1-c^i\right)}f\left(x,p,c\right)$$ My approximation is : $$\exp(x)=\underset{h,c\rightarrow+\infty}{\lim}q(x,h,c)$$
Desmos shows that it s indeed really close near $1$, and that for low $h,c$ it starts diverging afterwards (but this might be because of computational errors on huge numbers (?) ).
Is that a known approximation for the exponential ? If not, is it an approximation of the exponential at all ?
Additional question that sparked from the comments : How can the comportment for low $h,c$ be analyzed ?
If you wonder where that formula comes from, I can't give a full explanation (it's really experimental work) but you might be interested by one of my previous questions An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$.