title: Examples of Functors without Applicatives
body: <p>Are there any good examples of <code>Functor</code>s which are not <code>Applicative</code>s? By good, I'm seeking non-trivial (not <code>Const Void</code>) examples which don't need appeals to <code>undefined</code>. If there are none is there any method of proving that the space there is uninteresting?</p>
  
  <p>This is similar to <a href="https://stackoverflow.com/questions/7220436/good-examples-of-not-a-functor-functor-applicative-monad">Good examples of Not a Functor/Functor/Applicative/Monad?</a>, but it wasn't completely resolved there.</p>
  
  <p>As a follow-up question, are there any interesting examples of <code>Functor</code>s which might be left without <code>Applicative</code> instances due to having far too many non-canonical <code>Applicative</code> instances to be meaningful? For instance, "extended <code>Maybe</code>" is a bit boring</p>
  
  <pre><code>data MayB a = Jus a | Nothing1 | Nothing2 | Nothing3 | ...
  
  instance Applicative MayB where
    pure = Jus
    Jus f &lt;*&gt; Jus x = Jus (f x)
    Jus f &lt;*&gt; n     = n
    n     &lt;*&gt; Jus x = n
    n1    &lt;*&gt; n2    = methodOfResolvingNothingWhatsoever n1 n2
  </code></pre>
  
  <p>Are there examples where the variations of the <code>Applicative</code> instance are more material?</p>