Today I am a professor in computer science and still draw on it for examples in my advanced functional programming course. Just last week we did the loeb function, as an example of interesting use of Functor.

Loeb function: http://blog.sigfpe.com/2006/11/from-l-theorem-to-spreadsheet...

In addition to the free monad presented in this post, there is a variant, called the "freer" monad, based on the "bind" operation instead of the "join" operation:

    data Freer f a where
      Pure :: a -> Freer f a
      Bind :: f a -> (a -> Freer f b) -> Freer f b

When thinking of monads as giving the semantics of some computational strategy, it's easier to define them in terms of "bind" instead of "join." This way of defining monads is sometimes called a "Kleisli triple" because it is better suggestive of "Kleisli arrows," or functions of the signature `a -> m b`. The "bind" operation defines how to compose a monadic computation with its continuation, and from this perspective, the "freer" monad resembles an abstract syntax tree.

Originally, Eugenio Moggi proposed monads as a technique for specifying the denotational semantics of programming languages. All Java programs "really" happen in the IO + Either monads, because all Java programs may perform IO and throw exceptions. To my understanding, free monads are the monad that OCaml 5 runs in, because they give the semantics for effect handlers (or resumable exceptions).