All my favorite moments are probably 10 or more years old as I started to watch it in 6th grade (on Fridays and over the summers) and throughout high school but haven't in years. Much funnier back then, in my opinion. My Conan memories are:
The Masturbating Bear, Pimp-Bot 5000, Carl "Oldy" Olson, the Gaseous Weiner, staring contests with Andy (after the '96 Olympics with Bela Carolyi and Kerri Strug impersonators making out), driving the desk with Andy, the Max-on-Max channel, the subtitles controller changing the channel to Spice Channel porn because the show is boring, Conan searching for his old, original furniture and finding his couch on a farm mating with another couch...