Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by the regular representation) - that's an example of compact generation of modules for an algebra. But here we're asking about comodules for a coalgebra that's not dualizable so it's not immediately clear (to me).
This makes more specific my other question for any "nice" examples of non-compactly generated categories.