Note that energy conserving means that the material's output is <= 100%, not == 100%. A material that absorbs light and doesn't show it back is very common
I guess materials that absorb light in especially strange
What is not common though, is a material that can output more than incoming light.
Fluorescense is a great example of materials that can output more energy than its incoming energy. That's because they add an additional source of power other than incoming light (i.e. chemical reactions). Note that conservation of energy (as in the universe as a whole) is preserved, since they're just releasing energy they had been storing in some other form; but for the given time an observer was watching, he or she will have observed that outgoing light > incoming light.
I'm a bit curious about this. Since real-life materials are perfectly capable of absorbing light and then re-emitting it as energy other than visible light, what does it really mean to have a material that doesn't conserve energy? I guess materials that absorb light in especially strange, angle-sensitive ways are probably rare, but it seems plausible that some arrangement of microfacets could potentially be described by materials which are obviously "wrong."
Burning ashes could be another example. The extra light emitted by the ashes comes from heat.
In games, we just call those emissive materials and don't usually care about "correctness". Though this is starting to change (since Physically Based BRDFs + HDR + GI = We can use real life values, like sun's measured power, a bulb's measured power, etc)