So, I was trying to figure out what the max score on jeopardy is for a single day. what I did was account for the daily doubles at the very end with the lowest value category, (to save on the 1000's), Wagered everything in the daily doubles and final jeopardy. However I ended up with just a total of $approximately\;\$360000$ for a single day. However I read that the true possible win is $\$566,400$ total. Where am I messing up? And am I missing a rule or something?
I like this question, and I had to look something up for a moment. According to the Jeopardy Archive, daily doubles haven't occurred on the first row in years. But they can occur on the second row. So let's assume that all three daily doubles occur on the second row, meaning on questions of value $\$200$, $\$400$, and $\$400$. (As unlikely as this might be).
With this assumption, we naturally assume a player gets all the questions before doubling his or her money with the daily doubles.
During the first round, he or she will get $6$ questions worth $200, 600, 800, 1000$ and only $5$ worth $400$. This totals to $6\cdot (200 + 600 + 800 + 1000) + 5\cdot 400= 6\cdot(2600) + 2000 = 17600.$ Hitting the daily double, the player has $35200$ after the first round (a respectable amount of money already).
During the second round, he or she will get $6$ questions worth $400, 1200, 1600, 2000$ and only $4$ worth $800$. This totals to $2 \cdot 6 \cdot (200 + 600 + 800 + 1000) + 4 \cdot 800 = 2 \cdot 15600 + 3200= 34400$. Thus the player will have a total of $\$69600$ before the second set of double jeopardies. After them, the player will have $\$278400$.
Thus after the final round, assuming the player bets all of his or her money, he or she can end up with $\$556800$.
I presume that the small difference with my total and the total you suggest is that they let the daily doubles land on the smallest ones, instead of the second-smallest like I thought.