A stock reduced to a syrup is known as a glace. Glace is French for "glaze". The glace is used to flavor sauces. If you stop the cooking before it becomes a syrup, you have what is known as a demi-glace. If you cook it beyond the syrup phase it will probably burn. These are intended to be highly flavorful preparations.
If you reduce filtered broth all the way, you get portable soup. It dries down into a solid that looks a bit like leather. Because of the gelatin from the bones, portable soup is bendy and flexible. It was used in the 18th century as a portable food item, eg by soldiers and people traveling through the American wilderness. There's an excellent video by ...
Removing all the liquid out of stock is essentially how bouillon is made. If you managed to dehydrate the stock without burning it, you'll be left with a big disk of bouillon at the bottom of your pot, similar to the cubes you buy at the grocery.
If you leave it with just a little water, you'll have a gooey bouillon paste.