Melamine is also called particle board. It is basically hard pressed sawdust coated with a plastic resin coating (called Melamine). Or Particle board with veneer. I have also seen it called MDF (Medium Density Fiberboard)
All hardware/lumber stores should have full sized sheets, such as Home Depot, ROMA, Beaver Lumber, Cashway, Castle Building Center...and I can't think of any others. These may all be Canadian lumber stores..except the first, but pretty much anywhere you can get large sheets of plywood you should be able to get melamine.
You can use plywood sealed with a clear coated varnish such as Urethane or even a shallac. Just make sure the final coat is completely dry and you leave at least 24 hours in a well vented area to completely cure (or longer, check what it says on the can). Most indoor paints and varnishes that can be used around children, are safe for dragons.
Floor underlay can be used instead of plywood but you do need to build it on a frame. Below is a cage I had made for me for my first dragon (who unfortunately I had to give to a new home when I moved and wouldn't have my own place for a few months..)I have made similar cages myself but with nowhere near the same skill.
The problem with plywood however, is it can be even more expensive than melamine, if you want it sanded on both sides. Lower grade plywood is unfinished so an be very rough. I would suggest with plywood to get a non-pine version..as spruce may be too sappy and the pine toxins can irritate lizards. If completely sealed, this shouldn't be a problem but pine sap when heated could still seep out. So a hardwood based plywood (fir for instance) would be best. Also buying it finished/sanded on one side (making sure that sanded side is inside the cage (no worries about rough interior harming your dragon)) will make it easier to paint/seal so the inside has a nice finish.
Another thing about using plywood or even floor underlay if you use the frame method, is you want the inside to be extra sturdy, so more coats of a clear varnish either over the wood directly or as a finish over paint, will provide a hard coat finish to make clean up easier. Covering with a vinyl mat tac type sheet will help protect it too.
Particle board or melamine doesn't chip that much, if you work with it properly. Pre drill all holes and use screws instead of nails. A proper saw blade designed for cutting melamine will help chipping around cuts as well. Making sure not to place screws too close to edges or using too big of screws, or drill bits will help ensure less chance of splitting. I have used flat headed wood screws, size 8 that are 1 1/2 inch long. (which use the + or star shaped screw drivers..I think they are Phillips screws..but could be wrong) For predrilling i used 1/8 of an inch drill bits designed for wood.

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PHLdyPayne