You can definitely break even with N=13, assuming the unrealistic even distribution between 0 and 1, perhaps a bit earlier with a more sophisticated strategy.

Example strategy: Choose randomly the first time, the same side in all further throws. If the other side comes up before the 6th throw stop (6th throw because that's the earliest your expected winnings for the next throw don't increase by stopping if the other side comes up). Otherwise continue to N.

The probability of the same side coming up n times is 2/(n+1), the probability of the other side coming up exactly the n+1th time 2/(n^2 +3n +2).

Expected winnings each throw: 1st: -1$, 2nd: -1/3 $, 3rd: 0$ 4th: 1/10 $, 5th: 2/15 $, 6th-Nth: 1/7 $