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One example is the following - suppose that $M$ is a non-compact connected manifold of dimension $\geq 1$. Then the unbounded derived category of chain complexes of sheaves of abelian groups on $M$ has no compact objects other than zero. It is, however, well generated. This can be found in Neeman's paper "On the Derived Category of Sheaves on a Manifold".

Another example is suppose that $R$ is the ring $k\oplus V$ where $k$ is a field, $V$ is an infinite dimensional vector space and the ring structure is the unique one making $V$ a square zero ideal. Then the category $K(Proj\; R)$, the homotopy category of complexes of projective $R$-modules is not compactly generated. Again though it is well generated (even $\aleph_1$-compactly generated). In general this category need not be compactly generated if $R^{op}$ is not coherent.

One could also produce more examples along these lines I imagine by considering for instance the homotopy category of flat modules over a suitable ring.

These last two examples are very natural objects to study - I can provide more details on why if anyone is interested.

These are the most naturally occurring examples I know of off the top of my head.

Depending on what your motivation is

I'll try and think of a modular representation example if I might be able to say something more helpful?can although when I think about these things it is normally the stable category for modular reps of a finite group and these are always compactly generated.

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One example is the following - suppose that $M$ is a non-compact connected manifold of dimension $\geq 1$. Then the unbounded derived category of chain complexes of sheaves of abelian groups on $M$ has no compact objects other than zero. It is, however, well generated. This can be found in Neeman's paper "On the Derived Category of Sheaves on a Manifold".

Another example is suppose that $R$ is the ring $k\oplus V$ where $k$ is a field, $V$ is an infinite dimensional vector space and the ring structure is the unique one making $V$ a square zero ideal. Then the category $K(Proj\; R)$, the homotopy category of complexes of projective $R$-modules is not compactly generated. Again though it is well generated (even $\aleph_1$-compactly generated). In general this category need not be compactly generated if $R^{op}$ is not coherent.

One could also produce more examples along these lines I imagine by considering for instance the homotopy category of flat modules over a suitable ring.

These last two examples are very natural objects to study - I can provide more details on why if anyone is interested.

These are the most naturally occurring examples I know of off the top of my head.

Depending on what your motivation is I might be able to say something more helpful?