I like the following example because it is very close to the origins of homotopy theory (and also because it worked on it at the beginning of my career): proper homotopy theory. Objects are topological spaces, maps are proper maps, one can define proper homotopies via cylinders in the usual way, weak equivalences are proper homotopy equivalences, and cofibrations are proper maps satisfying the homotopy extension property. The 'cofibrant' part works as in a model category, this is a cofibration category, but there are very few fibrations. Moreover, the category is not complete, it doesn't even have a final object. This is because the map to a point $X\rightarrow *$ is not proper unless $X$ is compact. Proper homotopy theory is however very much developed on its own, and has been applied in many contexts.

I like the following example because it is very close to the origins of homotopy theory (and also because I worked on it at the beginning of my career): proper homotopy theory. Objects are topological spaces, maps are proper maps, one can define proper homotopies via cylinders in the usual way, weak equivalences are proper homotopy equivalences, and cofibrations are proper maps satisfying the homotopy extension property. The 'cofibrant' part works as in a model category, this is a cofibration category, but there are very few fibrations. Moreover, the category is not complete, it doesn't even have a final object. This is because the map to a point $X\rightarrow *$ is not proper unless $X$ is compact. Proper homotopy theory is however very much developed on its own, and has been applied in many contexts.