According to the second part of the definition of $ f, $ the value of $ f(8) $ depends on the value of $ f(7), $ which depends on the value of $ f(6), $ which depends on the value of $ f(5), $ which depends on the value of $ f(4), $ and $ f(4) = \sqrt{4} = 2 $ by the first part of the definition. So, we compute $ f(5), $ $ f(6), $ $ f(7), $ and $ f(8), $ in that order: \[\begin{aligned} f(5) &= (f(4))^2 - 1 = 2^2 - 1 = 3, \\ f(6) &= (f(5))^2 - 1 = 3^2 - 1 = 8, \\ f(7) &= (f(6))^2 - 1 = 8^2 - 1 = 63, \\ f(8) &= (f(7))^2 - 1 = 63^2 - 1 = \boxed{3968}.\end{aligned}\]