A well-understood dimensional analysis of the highest point of the projectile in a uniform gravitational field goes like
$$ [x_M] = [g^av_0^bm^c]\\ L = (\frac{L}{T^2})^a(\frac{L}{T})^bM^c $$
Equating the dimensions of both sides yields $$ a+b=1\\ -2a-b=0\\ c=0\\ $$
Then we have $x_M\sim\frac{v_0^2}{g}$
But how do we carry this analysis to nonuniform field, namely, $g^* = g\frac{R^2}{(x+R)^2}$, where R is the planetary radius.