All kinds. Gorgons, empousa, drakons, giants, titans, cyclops, hellhounds. Thanks to that whole being fictional thing here, you can actually see a list here. I've fought most of what's on there.
Most monsters at home are best killed with special metals - imperial gold or celestial bronze, mainly. And they eventually reform in Tarturus and make their way back, so killing isn't really the right word. I know from the classes that it's a little different here.