Model categories 13 — Find a model category you can prove is not cofibrantly generated.

Find a model category you can prove is not cofibrantly generated. This is just an annoyance, not a very significant problem, but it has been bugging me for a while. The obvious candidate for this is the simplest nontrivial model category, the one on chain complexes where weak equivalences are chain homotopy equivalences. Mike Cole is, so far as I know, the first to write down a desciption of this model category, though one certainly has the feeling that Quillen must have known about it. But how do you prove something is not cofibrantly generated?