The show is The Happiness Hut with Ken and Andy Breckman (renamed from 'Seven Second Delay with Ken and Andy'). The first episode I listened to was the "Stay on the Phone in Ten Words or Less" show, in which Ken and Andy had a contest in which each of them would call someone up, introduce themselves and the radio station, and would then try to keep the person they called on the line for as long as possible, with the added restriction that after the introduction they could say no more than ten words.
At the time I heard it it was the dumbest thing I had ever heard. Not only did it start out dumb, but it just kept getting dumber and dumber and dumber. Never would I have thought that a radio show that consisted mostly of dead air have so many twists and would be so amazingly hilarious.
Since then I have heard other shows which are even dumber, though. Another personal favorite is "Radio Word Scramble", in which Ken and Andy had kids call up and Andy gave them $5 to play a game that is so simple it is impossible to lose. Then, later in the show, Andy called the kids back to see if he could convince them to let him keep the money, with mixed results. Again, it started dumb and gets dumber and dumber and dumber. Also, enjoy the false start!
Finally, there's the "Comedy Monologue" episode. Andy happens to know one of the writers for the Craig Kilborn show, so during this episodes Ken and Andy solicited jokes from people calling in and pitched them to the Craig Kilborn writer. This isn't as dumb as the other shows, but it's still quite entertaining, and you get to hear Andy reject callers' ideas Hollywood style! Which is highly recommended.
If you have your own favorite Ken & Andy show, let me know!