Explaining $\cos^\infty$

What you have found is the unique, attractive fixed point of $\cos(x)$.

For more on this point and these terms, see this (MathWorld) and this (Wikipedia).

This is the unique real solution $r$ of $\cos(x) = x$.
For any $x \ne r$ we have $|\cos(x) - r| = \left|\int_{r}^x \sin(t)\ dt\right| < |x - r|$. This implies that $r$ is a global attractor for this iteration.

As already discussed in other threads:

What is the solution of cos(x)=x?

Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$

fhe fixed point of $\cos(x)$ (i.e. the Dottie number) can be written as a particular solution of Kepler equation, therefore it can be also expressed as:

$$ DottieNumber=\sum_{n=1}^\infty \frac{2J_n(n)}{n} \sin\left(\frac{\pi n}2\right)= 2\sum_{n=0}^\infty \left( \frac{J_{4n+1}(4n+1)}{4n+1} - \frac{J_{4n+3}(4n+3)}{4n+3}\right)$$

where $J_n(x)$ are Bessel functions.