Let $ a, b, c, d$ are written on board. So, $ a + 1 = mb$, m - rational; $ b + 1 = nc$ (it's impossible, that $ c + 1 = nb$, because $ (a + 1)/(c + 1)$ cannot be rational); $ c + 1 = kd$ ($ d + 1 = kc$ - impossible...), n,k - rational. It must be $ a + 1 = pc$ or $ c + 1 = pa$, p is rational. In first case we have b/c rational (impossible); in second - a/d rational. So, $ (a + 1)/d$ and $ (d + 1)/a$ are not rational. Contradiction. So, we can see, no more 3 numbers on a board. Answer: 3. For example: $ \sqrt2, \sqrt2 + 1, - \frac {\sqrt2 + 2}{2}$