Dig into the numbers behind Jeopardy winnings.
To win the maximum amount of money in a game of Jeopardy, you'd have to get pretty lucky: not only would you have to buzz in first and answer correctly on all 61 questions, you'd also have to pick the Daily Double questions last in each round, so you would have the highest amount of money to bet each time you picked them.
In addition, you would have to have the board work in your favor by hiding the Daily Doubles behind the lowest dollar amounts on the board. However, based on the results of the last two seasons, the Daily Double is never found in the first row of questions, so your Daily Doubles would have to be hidden behind a $400 question in the first round and $800 amounts in the second. So, all this being the case, you would sit with one question to go at the end of the first round with $17,600 (the total of all questions on a first-round board, except for the single $400 question remaining). You would choose that question, pretend to be surprised when it's revealed to be a Daily Double, then bet all of your money and double up to $35,200.
In the next round, you'd clear the board with the exception of two $800 questions, which you have again correctly surmised to be Daily Doubles. Adding your new question totals to your winnings from the first round, you now have $69,600. Doubling up twice, you finish the round with $278,400. The only question to go is Final Jeopardy, which you are presumably playing alone because neither of your opponents has any money. You double up one final time to finish with the maximum total of $556,800, obliterating Jeopardy master Ken Jennings's previous record of $75,000.