Maximal bid allowed is 24, so the minimal raise would quickly overlap the maximum. 23th round is the final possible round.
If it's overtime #3, the minimum bid is 4. If someone has a bid of 22, they can still be bid on in overtime #3, but it goes to 24, not 26.
It doesn't matter what the "minimum" bid is, people can still be bid on. There are plenty of people in the auction who have a bid of only 1 card. If overtime #4 happens because of a bid on one person, overtime #5 can happen because of a bid on a completely different person. One single bid on a unique person can cause a new overtime. Since there are 240 different possible bids that can happen on different players, then if there are 240 applications there can be 240 unique bids.
So yes, it is possible to have 240 overtimes.