TITLE: Categories which are not compactly generated
QUESTION [16 upvotes]: Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear any examples.. the ones I know are gotten as opposites of compactly generated categories or by slightly ad hoc geometric constructions)

REPLY [7 votes]: There is also, now a bunch of years later, a result of Hall, Neeman, Rydh that says over a field of char $p$,there are no non trivial compact objects in $D_{qc}(B\mathbb{G}_a)$. 
Moreover they prove for any algebraic stack ,$X$, which is quasi-compact, quasi-separated and "poorly stabilized" (roughly that there exists a point whose stabilizer contains a copy of $\mathbb{G}_a$) then $D_{qc}(X)$ is never compactly generated. 
https://arxiv.org/pdf/1405.1888.pdf