Here is my answer (essentially the same as @Pranay's answer).
The moral of the story is that e is hidden in every ball.
Taking
Taking OP's hint, we consider the four quantities:
$r=$ red surface area of the small piece
$y=$ yellow surface area of the small piece
$R=$ red surface area of the big piece
$Y=$ yellow surface area of the big piece
We will show that the following limit (note the elegant structure) equals $e$: $$L=\lim\limits_{r/R\to 0}\left(\frac{r}{y}\right)^\frac{R}{Y}$$
Assume the radius of the sphere is $1$.
Let $t$ be the thickness of the slice. Archimedes tells us $r=2\pi t$.
Let $a=\sqrt{1-(1-t)^2}=\sqrt{2t-t^2}$ be the radius of the yellow disks.
$L=\lim\limits_{r/R\to 0}\left(\frac{r}{y}\right)^\frac{R}{Y}$
$=\lim\limits_{t\to 0}\left(\frac{2\pi t}{\pi a^2}\right)^\frac{4\pi-r}{\pi a^2}$
$=\lim\limits_{t\to 0}\left(\frac{2}{2-t}\right)^{\frac{4-2t}{2t-t^2}}$
$=\lim\limits_{t\to 0}\left(\frac{2}{2-t}\right)^{\frac{2}{t}}$
$\overset{t=-\frac{2}{n}}{=}\lim\limits_{n\to \infty}\left(\frac{2}{2+\frac{2}{n}}\right)^{-n}$
$=\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$
$=\boxed{e}$
