Not knowing what type of soil or location, or what sort of weeds you are thinking about, I can only suggest 6" plus, but without membrane, be prepared that the shingle may will sink over time. Especially if you will be walking on it. If the weeds are perennial, they will still fight for light and in time may well overcome the shingle.

When we created our gravel paths, we made them about a foot deep. Nothing has ever come through them and once they bedded in were easy to walk on.