Actually, this issue does come up in square root algorithms if you are not careful. To take a simple example, suppose we work in base $12$ and we truncate using the floor function. Start with a bigger approximation to $\,\sqrt{2}\,$
which is
$\, 17/12. \,$ Using the Babylonian algorithm, the next approximation is
$\, (17/12 + 24/17)/2 = 577/408 = 16.9705.../12 \,$ and if you truncate this to
$\, 16/12, \,$ then the next approximation is $\, (16/12 + 24/16)/2 = 17/12 \,$ and we have a loop, unless we terminate the algorithm at this step since we are getting a bigger approximation. The exact behavior of the sequence of approximations to $\, \sqrt{x} \,$ depends on the exact details of how the arithmetic is done, but except for the first iteration, you should always terminate if this sequence stops decreasing.
This issue is a very general one. Suppose you have $\,\{x_n\},\,$ a sequence of real numbers given by a recurrence $\, x_{n+1} = f(x_n) \,$ that converges to a finite limit. Now replace the real numbers by $\,F,\,$ a finite set of numbers such as used in a computer system, fixed point or floating point. Then the computing function $\, f_F : F \to F \,$ corresponding to $\,f,\,$ no matter what the initial value or even if the original $\,f\,$ converges or not, must eventually loop, and in best cases the loop is a fixed point.
Just as a curiousity, this kind of loop happens in other bases, namely,
$\, \{2, 12, 70, 408, 2378, \dots\} \,$ which is OEIS sequence A001542 which relates them to continued fraction convergents to $\, \sqrt{2}. \,$