A very unexpected 333 that I found in math

While studying about dynamical systems theory for my ergodic theory course, I came across the extremely unexpected number 333. From Resonances and small divisors (Etienne Ghys, 2007), chapter 10 of Kolmogorov’s Heritage in Mathematics:

In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)]. 

In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].

333 is half the number of the devil. Moser probably cut half of a deal with the devil.

The heart and soul of elliptic, flat (Euclidean), and hyperbolic spaces

... is constant, polynomial, and exponential.

Think about building cellphone towers in a Euclidean plane. First, we choose the central station arbitrarily at point $O$, and call it the "origin". To cover all space that's within a distance $R$ to the origin, we need $O(R^2)$ stations.

This is polynomial growth.

Then think about building cellphone towers on a spherical surface, on a little asteroid. Then it's clear that there's only a constant number of towers needed no matter how big $R$ is chosen, since the whole surface has finite area. So we need $O(1)$ stations.

This is constant "growth".

Then, if you know how to play in hyperbolic space (if not, go play Hyperrogue!), you'd know that the number of stations grows as $O(\exp{(CR)})$, where $C=(-K)^{-1/2}$, and $K$ is the Gaussian curvature of the plane. This can be derived by using the formula of the area of a hyperbolic circle of radius $R$:
$$\frac{4\pi}{-K} \sinh^2 \frac{r}{2\sqrt{-K}} $$

We normalize $K = -1$ to get the growth rate $O(\exp{R})$.