I'm pretty sure you would need the serial number to upgrade. Not when you send in the discs (if you're taking advantage of the
program), but when you get the new disc to reinstall.
However, if you bought it online and call customer service (and you have all of your original order information like order numbers and whatnot), you might find a kind soul who is willing to look up your information. I had to fax them a bunch of stuff, and I'm sure it helped my personal tale of woe that my house had burned down. But I had the original email from Apple saying that my order was shipping, so I could give them a ton of information for them to look stuff up.
