I'm a front-end manager for Target and I often get returns that don't have a receipt and were purchased with cash. At target we can just scan your license and use it as a frame of reference so we can know how much you've returned without a receipt (There's a limit of 70$-75$ per year you can have returned on a license). The problem of course with this is that it has to be put towards store credit so there's no way to get your cash back if you don't have a receipt. If you've reached the refund limit on your license you are restricted to doing an exchange for the amount of your return in a similar department, but I just tell most people to come back with their Boyfriends/Girlfriends/Spouses/SO/friends license and we can return it with that.

We can also look it up off of a credit/debit/charge card if the purchase was made within a certain time frame (if the guest has received a new card but retains the same account we can't look it up since the card is gone, and our system isn't allowed to see into accounts). Generally I won't accept returns without receipts if the item is clothing or shoes that have been worn/smell awful/are badly stained. The only other exception is electronics, electronics either need to be returned off of a receipt or card, if you don't have that then you are screwed.

I dislike the attitude of the OP just because even if they know that information, looking up a receipt in our system with even the slightest variation in time frame is a bitch. If a guest gave me a window between 1 and 1:45 for when they checked out, and they couldn't remember their register number, then I would have to look through hundreds of receipts in our system.

TL;DR Returns are hard, keep your receipt, don't be a dick and expect you can return everything without one.