Not sure what kind of used shops you have over there, but the ones in the US often have professional polishing machines that may do a better job. They'd charge a few bucks, but would be cheaper than completely replacing it.
Damn. How hot was the surface that those DVD's were on? I stack discs all the time and have never seen anything like that.
^ Me either. That looks seriously impacted into the plastic. Not sure anything can be done, save a VERY rigorous polishing to sheer off the layers.
"Now, Shelley. You know that Jesus cries every time you ruin your mother's Star Trek DVD." Honesty is STILL the best policy. Sheldon, your best bet is your local version of GameStop gaming store. Or, if one still exists in your area... a DVD rental store. They have the means to polish out damaged discs so they can re-rent them again. That disc appears to have been exposed to heat and pressure to create such an imprint. Probably from books laid on the discs, perhaps on top of the DVD player which created the heat and pressure. If the polishing at the rental place does not work, you need to replace the disc AFTER apologizing to your mother. If it was MY DVD of Star Trek IV: The Voyage Home, there would be hell to pay! The Final Frontier, meh, not so much.