We can have $N=3$ in the following way: teams $8$ through $15$ lose to everyone with a lower number than them, and for $i\in\{1, 2, 3, 4, 5, 6, 7\}$, team $i$ loses only to teams $i+1, i+2$ and $i+3\mod 7$ where the $\mod7$ sends it to a number in $\{1, 2, 3, 4, 5, 6, 7\}$. Clearly teams 1 through 7 qualify. Now suppose we have at least 7 teams with two losses or fewer. That means total they have at least $7\cdot12=84$ wins. There are only a total of $\frac{15\cdot14}{2}=105$ games, so there are at most $105-84=21$ games not won by these 7 teams. But the other 8 teams play among themselves in $\frac{8\cdot7}{2}=28$ games, and there's a winner from each of them. This is a contradiction. So $N=3$ is the minimum.