Solid Woods cut up to 1/2″ will cut, the denser the better (white oak cuts real nice for example).
Plywoods, MDF and Masonite up to 3/8″ will cut, possibly thicker depending on their glue content.
Acrylic of any color or mirrored up to 1/2″ will cut. Generally cast acrylic is better for engraving and etching and extruded is better for cutting as its edge finish is clearer.
PETG Plastic sheets work great as resilient stencils.
Fabric, both natural and synthetic can be cut. All fabrics will seal their edges to some degree but synthetics are great for laser cutting as their edges seal nicely with no fraying.
Leather, as thick as you can find it. The material will have a strong smell for a few days but it eventually dissipates. Since every piece of leather varies in density and chemical composition, it’s best to test a piece first.
Paper Products, such as cardboard, cardstock, chipboard, museum board etc. Depending on the makeup, can be cut in stacks to reduce cutting time and cost. To achieve a clean stacked cut, the paper should be relatively porous so the vacuum system can pull the paper flat and disallow smoke to infiltrate between the layers.
Stone can be engraved for lithographic purposes or others as well.
We can’t cut anything that contains vinyl, which includes PVC as this creates Hydrogen Chloride gas that is highly toxic and eats our mirrors and lenses and eyeballs which we would like to hold on to for as long as possible.
We can’t cut metals at this time, though we are in the process of installing a new optic that may allow us to cut thin sheet metal.