Existence, Uniqueness & Bifurcations · Hard problem

For $y' = y^2$, $y(0) = 1$, the Picard–Lindelöf theorem on the rectangle $|t| \le a$, $|y - 1| \le b$ guarantees a solution on $|t| \le h$ with $h = \min\!\left(a, \dfrac{b}{M}\right)$, where $M = \max |f|$ on the rectangle. Taking $a$ as large as you like and choosing $b > 0$ optimally, what is the largest $h$ the theorem guarantees?