Since it is very much possible to have functors with $F_0 \dashv F_1 \dashv F_2$ or even $F_0 \dashv F_1 \dashv F_2 \dashv F_3$, it begs the question:
Can there be an example of infinitely many, non-repeating mutually adjoint functors?
Obviously, "non-repeating" is important, since $\operatorname{id} \dashv \operatorname{id} \dashv \dots$ would otherwise of course be a boring example (and every other pair of pseudo-invers functors would be too).