John K Jordan
However far you go down, it is recommended to put a layer of geotextile on the dirt, then the gravel, then whatever topping you desire. This will even stand up to the destructive hooves of cows and horses, even when the ground gets wet. A friend of mine had a winter "muddy" area by his barn where the cows would sink almost 2' into and even his wife with waders got stuck and had to be rescued. In the summer he dug down, spread the geotextile then the gravel. It's been almost like concrete since.
I buy geotextile in rolls 15' wide and I forget how long but even Amazon sells smaller sizes. Or ask a local farmer or contractor if you need only a small piece.