Let $X$ denote a particular 512-bit sequence.
Let $A,B,C$ denote sequences of bits such that:
$$len(X || A) = len(X || B) = len(X || C) = 1600$$
$$\operatorname{Keccak-}f(X||A) = X||B$$ $$\operatorname{Keccak-}f(X||B) = X||C$$ $$\operatorname{Keccak-}f(X||C) = X||A$$
then we have a loop, so the output of SHAKE (if my understanding of how the sponge construction works is correct and if $X||A$ happens to be the state which was output after xoring the final block of input) is nothing but an infinite concatenation of the same 512-bit block. Is it possible?