Watched Conan since '93.
I'll check out Letterman too, but I'm more of an old-school Dave fan, from when he was a young upstart on NBC (where Conan is now). In those days, the show was a lot funnier. Andy Kaufman would often show up. Not because he was promoting anything, Dave just liked having him. Some classic Andy moments:
A dissheveled Andy, deadly serious, saying he's out of work and in need of money. He walked around the audience shaking a paper cup.
Andy, wearing a diaper and turban, swallowing a sword, then slapping on a fake mustache and singing a sincere version of Slim Whitman's "Rose-Marie."
Andy bringing his real-life mom and dad on the show, and apologizing to them for being a bad child when he was little, and telling them how much he loved them.
Andy's adopted "children" - a group of adult African-American men.
And of course, the notorious fight between Andy and wrestler Jerry Lawler.
Chris Elliott would also frequently appear, as would the recently-departed Larry "Bud" Melman.
Dave would wear a full-body suit made out of Velcro and jump against a wall, where he stuck.
He once did a show via "monkey-cam" - a camera strapped to a monkey.
He'd often go onto the roof of the studio and drop watermellons, water balloons, glassware, and other breakable objects onto the street below, just because it was "really neat" to see things break.
Yeah, Dave's NBC show was a lot of fun.