I found an interesting example of the lazy State monad that uses "head recursion".
I tried to adapt the example for the lazy Writer monad but it proved difficult. If you naively consume the head of the infinite accumulator, it will hang, even with the lazy version of the monad:
The trick is to use
Maybe I suffer from a failure of imagination, but I can't think of any practical problem in which the lazy Writer monad offers an advantage over the strict version.