The bullring of Almadén is famous for being the only hexagonal bullring in the world. During the off-season, it's looked over by an adjacent hotel-restaurant who will let you enter and explore the bull ring for free. One nice thing is that you'll typically be the only one visiting when bullfights aren't going on, so you have the chance to enjoy the stately architecture, wooden roofs, and white-washed arches without any disturbances. This should definitely be on your to-do list in Almadén!
The Royal Miners' Hospital in Almadén is one of the more interesting things to visit in Almadén, after the actual ancient mercury mines themselves. The hospital is a stately villa that has been renovated and converted into a museum which highlights the history of mercury mining in Almadén, the ailments that mercury miners faced, the controversy over the actual toxicity of mercury, and the importance the mine had for the town (let's suffice to say that it was the lifeblood of the entire area).
The hospital also has a nice courtyard in the back that has a Moorish-style fountain and plenty of flowering plants. The hospital also served as a jail and they still have the grim dungeons which you can visit. Again, it's not the most exciting things on the planet, but it's worth a quick visit if you're in Almadén to help you understand the history of the village and its people.