Set a calculator to radians and press cos repeatedly, starting anywhere. The display stops moving at
Why the button always wins: after one press you are in $[-1,1]$, after two in $[\cos 1, 1]$. On that interval the derivative satisfies $|{-\sin x}| \le \sin 1 \approx 0.8415 < 1$, and cosine maps the interval into itself. The Banach fixed-point theorem then guarantees a unique fixed point and geometric convergence toward it — every press multiplies the error by roughly $|\sin(0.739\ldots)| \approx 0.674$ in the limit, a digit of accuracy every three presses or so.1
1
Impatient readers switch to Newton on $f(x)=x-\cos x$:
$$x \;\mapsto\; x-\frac{x-\cos x}{1+\sin x},$$which doubles the correct digits per step instead.
Two remarks close the file. First, the fixed point is global: cosine is the rare button whose iteration converges from every real start, which is why the trick survives on any calculator. Second, the number is transcendental — by Lindemann–Weierstrass, the cosine of a nonzero algebraic number is transcendental, so an algebraic fixed point would contradict itself. A number defined by one keystroke, and no polynomial will ever own it.