What about the Hollywood Squares? Is it possible for a perfect game to happen?
I take a stab at Finders Keepers. All hidden pictures found, all hidden items in the room searches are found, the room to room romp are successfully completed.
Yes, there's definitely a concept of a perfect show on Squares. Just answer all questions correctly, find the Secret Square(s), and don't make any mistakes in the bonus round, including choosing the correct key on some versions. Probably Plinko-type odds of it happening, but it's certainly possible, and a more realistic scenario is to simply have the Secret Square(s) won as well as the bonus round if necessary.
Here are a few more:
Hot Potato: Answer all seven answers correctly on each maingame question, plus win the bonus round without making a mistake
Body Language: Get all five words in each puzzle, answer every puzzle correctly, and then win $10,000 in the bonus round
Nick Arcade: Beat every Video Challenge, beat the Video Zone, answer every Pop Quiz/Video Puzzle/Goal question correctly, and also avoid the enemy
Get The Picture: Answer every question correctly, solve each puzzle with only one square showing, beat every Power Surge, and beat Mega Memory
GUTS: 1,925 points
Make The Grade: "Graduate" from the game quick enough so not only you win (without missing any questions and winning any Fire Drill), but also having enough time to play and beat the Honors Round and the University Round
Figure It Out (happened very often): The panel stumps both contestants
Think Fast (individually): Beat all stunts, solve at least one Brain Bender, and win the bonus round
Debt: Like Jeopardy!, every question is correctly answered and/or rebounded (in General Debt), along with winning the Get Out Of Debt round perfectly by answering ten questions in 60 seconds and then taking the doubler and answering the question. Due to time constraints, the General Debt board was never cleared, so that isn't a qualification, except if the Debtonator isn't found before the round ends