I believe I found a simple sample for a ccc, based on the answers from Tom Leinster, Finn Lawler and John Bourke, and http://opus.bath.ac.uk/23104[1]

I also used the fact I found in Moggi's "Computational lambda-calculus and monads" - that a category should be [well-pointed][2].

Take category **2** (two objects, three arrows), and a topos Set<sup>**2**</sup>. This topos is obviously not well-pointed, so we can proceed. Take a monad similar to the one described in http://opus.bath.ac.uk/23104/[1]. Namely, (a: a<sub>0</sub> &rarr; a<sub>1</sub>) maps to a + (0 &rarr; a<sub>1</sub>), with obvious unit and multiplication.
Now this monad is obviously not strong.

I believe this kind of topos would be a good testing area for the favorite Haskell constructs. Some of them won't hold, I believe.

Now I wonder... can we prove that if all monads over a topos are strong, then the topos is Boolean? Will post it in another question.

  [1]: http://opus.bath.ac.uk/23104/
  [2]: http://en.wikipedia.org/wiki/Well-pointed_category