Ohhhhhhhhhhh, @fisherlady !!!
You are the bestest person to ask! You know I am ... Not the sharpest tool.you have experience with gravel in the run, and you have seen my set up.
Tomorrow I will be removing the chicken wire from the run. I have a landscaping supply yard behind me. There are a few others in town as well if he is unable to help me with such short notice.
What grade/size gravel do I need? How thick do I need it to be? If I use the Rototiller to loosen up the soil, it comes out almost even with the top of the 2x4s edging the run. How much should I (try to) remove?
If you like math equations, the run is 6 x 24'.
What I used is a "crushed limestone" which is 1/4 to 1/2 Inch. The base layer needs dug down to relatively solid ground, which is probably a couple of inches down in your run which should get you to where your previous gravel has been pushed to.
To cover an area 6x24x2inches thick will take about 0.9 cubic yards, crushed limestone weighs about 2800 lbs/cubic yard so to do a 2" layer would take about 2600 lbs. (that allows a bit of extra)
If you just get 1.5 ton it should allow you enough extra to also do the area outside of the run between the broody area door and the run door. I also keep a garbage can worth of it stored somewhere here to fill in low spots or holes they dig into it around the edges.
Thank you.
I was just reading some samples for some "inappropriate children's books"! 
