Don't have the box, don't have the CD, didn't register it?
I'm not sure there is a way to retrieve your LR 4 serial number in this case.
I do recommend that you develop a method of storing your serial numbers (not just for Lightroom but all software) that will survive computer death and other possible other catastrophes so this doesn't happen again in the future.
OK, I understand it was a big mistake on my part to not register or record the serial somewhere, combined with an untimely computer death.... but now that I've paid the $79.00, downloaded and installed the upgrade version, how can I get the full version? I'll pay the difference, that's not an issue. The online help chat does not work and there are no phone numbers listed to call for support. It seems ridiculous that my only option is to pay the full price and download again. Did I just set $79.00 on fire?
Sorry, we can't help you in this forum, we don't work for Adobe. You'll have to keep trying to contact them.