I had an original Mike Dirnt P-Bass well over a decade ago. That thing had the thickest neck I've ever held. My guess is that it is the same or similar to the 50's C-neck mentioned above.
I find the MIM 50's classic necks to be quite thin front to back. Sure, the 1.75" nut make the fretboard a little wider.
70's P bass neck with a cello fingerboard grafted on. The only thing thicker is an upright. It laughs at anyone who asks about silly things like "truss rods" or "neck relief".
Not Fender but IME the thickest I have found are Warmoth standard P , pretty damn thick , I dare say too thick for me and Pittman , though more known for his guitars his basses have some serious meat to them , he will also carve what you want . This is a Pittman V to C which is my fav Pittman Bass Build