A calculus-free proof that the volume $V_n$ of the unit $n$-sphere goes to 0 faster than any exponential. Equivalently, the volume $r^nV_n$ of the sphere of radius $r$ goes to 0 for every $r$. It is inspired by the intuitive answers about concentration of measure.
Claim. For any $0 < h < 1$,$$V_n \le 2h V_{n-1} + (1-h^2)^{n/2} V_n.$$
Proof. Remove a slab from the middle of the $n$-ball with thickness $2h$, and bring together the remaining slices to make a lens shape. The radius of the equator of this lens is $\sqrt{1-h^2}$, and it clearly fits inside of an $n$-ball of that radius.
Proof of main result. Rearrange the claim as a volume relation between adjacent dimensions:$$V_n \le \frac{2h}{1-(1-h^2)^{n/2}} V_{n-1}.$$For every $h$, the factor on the right is eventually close to $2h$, qed. In particular, if we take $h = 1/3$, then by the time that $n \ge 19$, the volume has turned around and is decreasing.