User blog comment:DWither/Doubt about busy beaver/@comment-5529393-20180904212819/@comment-5529393-20180905014340

Ah, I see... so you're just looking at if you repeat a sequence of states and inputs (which will lead to the same output and movement instructions). Unfortunately, there is no reason why a Turing machine should ever repeat such a sequence forever either. A simple example is a Turing machine that prints a 1 and moves right, then prints a 1 and moves left until it reaches a 0, then prints a 1 and move right until it reaches a 0, then prints a 1 and moves left until it reaches a 0, etc.