What I missed is the requirement that you ignore the known upper bound, even though it's right there in the code, and so you can't use the move counter technique to achieve O(1)/O(n), you have to use the cycle-finding algorithm, which is slower but at least still not quadratic.
"Your code should not presume anything about the gameboard’s size
or contents, only that it is given an arrow every time though the
while loop."
However, in my experience 0% of interviewees I've ever sat with, given such a function, ES6Fiddle and nothing else, would actually be able to implement Floyd's cycle finding from memory. Unless of course the recruiter prompted them to study it the night before.
If you're adverse to providing full internet access during interviews (which I always do) then at least you would have to provide an excerpt from a relevant textbook or Wikipedia to set them on the right path.
It's an extension of your idea of counting the total number of visited positions. What I like most about it is that it can work with streamed data.