I know that Limewire does not give refunds, because they have a free version for people to test out. Since they is no sign-in or anything, who would stop people from just saying that their Limewire does not work and demand money back.

This is of course if you bought it from limewire.com if not, then you have been scammed and should contact your credit card company for a refund. There are a lot of scam sites promising people millions of legal free songs, Lifetime membership etc. They are selling a product that in no way belongs to them in the first place.