jmc

For $x<a,$ the graph of $y=f(x)$ is the same as the graph of $y=ax+2a,$ which is a line with slope $a$ and which passes through the point $(a,a^2+2a).$ For $x\ge a,$ the graph of $y=f(x)$ is the same as the graph of $y=a{x}^2,$ which is a parabola passing through the point $(a,a^3).$

Notice that the parabola only ever takes nonnegative values. Therefore, the line portion of the graph must have positive slope, because it must intersect horizontal lines which lie below the $x-$axis. Thus, $a>0.$

For $a>0,$ the line portion of the graph passes through all horizontal lines with height less than or equal to $a^2+2a,$ and the parabola portion of the graph passes through all horizontal lines with height greater than or equal to $a^3.$ Therefore, all horizontal lines are covered if and only if $$a^2+2a\ge a^3.$$Since $a>0,$ we can divide by $a$ to get $$a+2\ge a^2,$$so $0\ge a^2-a-2=(a-2)(a+1).$ This means that $-1\le a\le 2,$ so the greatest possible value of $a$ is $\boxed{2}.$

The graph of $y=f(x)$ for $a=2$ is shown below (not to scale); note how the parabola and line meet at one point: