Let $q$ be a prime $q\equiv 3 \pmod 4$, and consider the finite field $F_{q^2}$.
We take the sequence starting with some $x_0$ and with
$x_{i+1} =\sqrt{\frac{x_i+x_i^{-1}}{2}}$ for $i\geq 0$.
We understand that this terminates if the element in the radicand is not a quadratic residue, so the square root cannot be taken.
Surprisingly, if I start with $x_0=\sqrt{\sqrt{2}}$, this never terminates (I have tested many cases). Does this have an explanation?
