Does a closed loop system with a reference of its previous iteration always reach a point in which it adventually loops, even if that loop is stable?
As is usual with these sort of questions, the answer is "it depends".

For example, suppose
$$
F(x) = x
$$
then the loop is immediate.

For example,  suppose
$$
F(x) = x/2
$$
then the loop never occurs unless $x = 0$.

The same will be true for more complex $F$s. 

For example
$$
F(x) = \mathbf{A} x
$$
where $\mathbf{A}$ is a matrix. If $\mathbf{A}$ is a rotation matrix of a certain type (so that it rotates by $360/n$ degrees where $n$ is an integer) then the period will be $n$. If $\mathbf{A}$ is something else, then all bets are off.