Page's guitars have super-thin necks apparently. Like, they were actually shaved. But, he's a pretty tiny dude, probably has small mitts, so it seems to fit.
I don't really have a super strong preference. My meatiest neck is the one on my G&L ASAT, it's right about 1" at the thickest spot. My LP Trad and my PRS Mira are both just a bit smaller than that and my LP Custom has one of those tapered jobs they did in the 70's, from quite thin at the nut to really huge at the body. My G&L Legacy is your basic mid-size C profile bolt-on type of neck that probably 90% of all Strats have. My other Gibsons (SG and Firebird) have slim-tapers and I like them just fine, vive la difference, you know.