title: Good examples of Not a Functor/Functor/Applicative/Monad?
body: <p>While explaining to someone what a type class X is I struggle to find good examples of data structures which are exactly X.</p>
  
  <p>So, I request examples for:</p>
  
  <ul>
  <li>A type constructor which is not a Functor.</li>
  <li>A type constructor which is a Functor, but not Applicative.</li>
  <li>A type constructor which is an Applicative, but is not a Monad.</li>
  <li>A type constructor which is a Monad.</li>
  </ul>
  
  <p>I think there are plenty examples of Monad everywhere, but a good example of Monad with some relation to previous examples could complete the picture.</p>
  
  <p>I look for examples which would be similar to each other, differing only in aspects important for belonging to the particular type class.</p>
  
  <p>If one could manage to sneak up an example of Arrow somewhere in this hierarchy (is it between Applicative and Monad?), that would be great too!</p>