Farthest Orbit around the Earth? What is the farthest orbit that can be achieved around the Earth? How fast is that?



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I ask because of this question: https://worldbuilding.stackexchange.com/questions/104593/can-you-get-a-orbital-mass-driver-to-work-this-way

Is there an orbit exist where 2 objects going in opposite directions can intersect repeatedly with minor course corrections?
The farthest Orbital Point from Earth is regulated by the fact that at what point the gravitational pull of the Sun overcomes the gravitational pull of the Earth.

I found a post for this on Quora

This was the most up voted answer by user Paul Olaru on Quora:-


  Approximately 1.5 million kilometers. It is limited by the point at which the influence of the sun's gravity becomes sufficiently stronger than Earth's that any object at that distance would become unbound from Earth &amp; end up in independent orbit around then sun.
  
  That limit applies in general to any system of two bodies, and is called the Hill sphere. It's not actually perfectly spherical, but close enough for most practical purposes, and lies between the L1 &amp; L2 Lagrange points. For circular orbits, the radius of the Hill sphere for any planet or moon can be calculated from
  
  r = a*(m/(3M))^(1/3)
  
  Where r is the radius of the Hill sphere, m is the mass of the planet or moon, M is the mass of the sun a planet orbits, or the planet that a moon orbits, and a is the distance from the sun (or planet) to its planet (or moon).