I've always preferred fatter necks, but I can live with Fender's "modern C" shape, anything thinner than that isn't for me.
Having said that, I agree with Simon D: I've realised recently that perhaps it's the shape, as much as the size, which matters. I don't like V necks, and I also don't like flat-backed, square-shouldered D shapes, a rounded shape works best for me. In fact, I like the cross-section to be pretty much a semicircle, so the depth is half the width. So, the best shapes for me are the Gibson 59 LP or Rounded SG shapes - they're relatively shallow at the nut but get a lot deeper as you move up the neck.
I'm not so keen on necks which are the same depth all the way along, whether they're thin or fat - I was expecting to like the big "soft V" (which is more of a U, really) on the Baja Tele, but in fact there's just too much wood around the first few frets and I struggle with it. Now I'm sure I wouldn't get on with a Warmoth Fatback or Boatneck.
(God knows what Afghan Dave is going to make of this thread.... :? )