No, it's not The Simpsons or even the Today Show. The longest running continuously aired television show in the history of this continent actually started on radio in 1945, and graduated to the tube two years later. It's NBC's Meet The Press and there hasn't really been a week when it hasn't been on the air on that network. It's an incredible accomplishment, especially considering how volatile and changeable the industry has become. 

The Simpsons notwithstanding, I doubt any other show will ever reach this milestone - and it shows no sign of ending any time soon. Pretty amazing. 

‘Meet the Press’ Turns 75: Where Are Its Former Moderators Now?