cheap guitars with big necks

Epiphone 56 Les Paul Standard Pro, Epiphone 55 Les Paul Custom, Epiphone Brent Hinds Flying V Custom. As for Fenders, check out the 50's Baja Tele.
Of course, if you consider the neck on the Traditional Pro fat, you'll probably think any Epiphone or Gibson neck is fat. That guitar has a slim taper neck, too thin for me.

 

I dunno, I think maybe they accidentally put a bigger neck on some of them. Specs for the Epi Les Paul Standard I used to have (built in 2013) said it should have had a D-shape slim taper neck, but it was definitely a fatter rounded C shape, more like the Gibson 50's neck. I haven't found another one yet that was the same model with a neck as fat as mine was. Might be the same thing with your Traditional Pro, all of them I've played (both Traditional Pro and Traditional Pro II) had the flat back D shaped slim taper neck, nothing like you're describing at all.

 

PRS SE 245s are pretty chunky.

 

92 peavey predator

 

Measure or we are all guessing as to what big means to any one person.
Width, thickness and shape all count.

 

The Squire J. Mascis Jazzmaster has a beefy neck.

 

G&L offers a variety of neck sizes, both in overall "girth" and nut width. It's the way to go to get a "custom" neck on a Fender style guitar for not much money. You can find them all day for $700 and up (before shipping) for used USA models, and since most people keep the "birth certificate" of the guitar, you can generally find out what size neck it is before buying.

 

I find this hard to believe. I have a Bernie Marsden model, which is supposedly thicker than the 245, but it is noticeably less chunky than my R9.

 

Hamer Vector XTs are like 2x4s

 

lamenlovinit beat me to recommending G&L.

 

Baja 50's is pretty chunky. Also USACG's standard 7 1/4 to 9 1/5 soft V to C compound necks. Not exactly cheap but not really expensive either.

 

I second the Squier J Mascis JM. The neck size and shape was spot on to an R9 I had.

 

I just found a brand new 2015 P245 quilted 10 top and grabbed it. It has a stellar '59 shaped neck. It feels as good as all the Gibson '59 neck guitars I've owned, and the built of it is simply stupid perfect. Small shoulders, beautiful fullness and feel.

 

Putting bright humbuckers in a PRS SE, is what got me to switch from Fender and G&L single coils. Today, my only single coils are P90's or Lipstick pickups.

 

The Hendrix Strat
Ibanez Fireman (Paul Gilbert model)
Epiphone '55 Custom
The G&L Legacy Tribute

 

Probably the PRS SE models that are "wide fat" are going to be your best bet. They are modeled after a '59 neck, they don't cost that much and the quality is usually very good.