I don't think that's true; there are computations that take infinite steps but never converge or repeat, like the mandelbrot set.
Looping for millions or billions of steps is absolutely normal in traditional algorithms. We know from complexity theory that some computations require a minimum number of steps. More depth is just more room for computation.
These are not deterministic functions or systems that have infinite precision. See the "should I drive or walk my car to the car wash", or any of the other logic riddle problems, for examples of a statistical attractors.
No, not hold more information, perform longer computations.
E.g. if you want to solve sudokus, you will need more and more loops for larger grids. There is no shortcut.