I've replaced a number of things through Logitech over the years.
I think the total now is like 4 keyboard/mouse combos, a USB headset and a speaker set.
9 times out of 10 they will only ask you for your serial number. Sometimes the rep will walk me through some troubleshooting techniques (hey whatever, they're just doing their job...most of the time I just explain to them politely that I'm a very knowledgeable PC user and I have tried all the common solutions and have even gone through some browsing sessions to look for common solutions and troubleshooting techniques)
@Vex -
I have replaced 2 KB/Mouse combos through them where they have requested my sales slip. It's my belief that they only do this with outdated or discontinued hardware due to the fact there's always people out there looking to get stuff for free (i.e. a free upgrade when the current hardware is not capable of warranty replacement) Every item I have ever replaced through Logitech that wasn't discontinued has been replaced in timely fashion with new merchandise (no refurbs) without any questions asked other than the serial number. The only item I have ever had to ship back was the USB headset because the problems with it were pretty small and honestly I doubt other manufacturers would give me less of a hassle about returning it if they even let me return it at all (mute button light burnt out and due to the location of the mute button, this really was annoying to me). Even then, they shipped the replacement first and just had me hand off the box to the UPS guy, all postage paid for by them.
Warranty replacement and service is definitely IMO one of Logitech's strong points.