The positive integers $x$ and $y$ satisfy $xy=x+y+3$. The greatest possible value of $x+y$ is:

Rewrite as $xy-x-y+1=4$, i.e. $(x-1)(y-1)=4$. The positive-integer factor pairs of 4 give $(x,y)=$ (2, 5), (3, 3) or (5, 2). The sums are 7, 6 and 7, so the greatest is 7. Check (2, 5): 10 = 2 + 5 + 3.