One can easily prove that if the attack continues, then the difference of the largest integer and the smallest integer strictly decreases, hence there are at most $N$ attacks. The maximum can be achieved when $1,N,N+1$.

We need to assume that the largest two squirrels never go below the smallest one because this can make things go negative. The problem becomes quite straightforward. We have that the largest always decreases,thus we get a bound of $N$. The situation where this could arise is when the squirrels have the integers $1,N,N+1$

this is now modified to the statement we got in the camp. The original statement and the statement in USJL's file give you the same answer, but the official statement is stronger. The statement in USJL's file is still "Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make?" for now(2/3)