Answer
The least initial mechanical energy required at launch is $~~E = 0$
Work Step by Step
Let $M_E$ be the mass of the Earth.
Let $m$ be the mass of the projectile.
The minimum escape speed is $v = \sqrt{\frac{2GM_E}{R_E}}$
We can find the least initial mechanical energy required at launch:
$E = K+U$
$E = \frac{1}{2}mv^2-\frac{GM_E~m}{R_E}$
$E = \frac{1}{2}m(\sqrt{\frac{2GM_E}{R_E}})^2-\frac{GM_E~m}{R_E}$
$E = \frac{1}{2}m(\frac{2GM_E}{R_E})-\frac{GM_E~m}{R_E}$
$E = \frac{GM_E~m}{R_E}-\frac{GM_E~m}{R_E}$
$E = 0$
The least initial mechanical energy required at launch is $~~E = 0$
