I thought if you weren’t making money you could tow what you want to?
Like many other things, it depends. Largely on location. Some areas will not let you tow many single car hauler trailers personally without a CDL, some areas will let you run a OTR truck and trailer as long as it's strictly personal. Typically falling under RV regulations, which is how it's treated here in Arizona.
Want some fear in your thoughts? Here in AZ I can legally go buy myself a tandem axle tractor, slap a 53ft trailer behind it, load it up with all my junk, and hit the highway on my regular ol' driver's license with zero training on how to drive a big truck.
That said, what defines "making money" is one of those gray areas. Examples commonly given are things like - hauling your antique AG tractor/antique car/horse/etc to a show, and the 1st place trophy comes with a $100 cash award as well? You've just hauled that thing "for profit", and need a CDL for it. Help your buddy move all his junk to a new house, and he throws you gas money? You've just hauled "for profit". Of course, actually enforcing that is a whole 'nother story. On the other hand, I've heard stories of guys getting hassled by DOT officers in big trucks who didn't believe anyone would have such a thing for personal use, though that's slowly changing due to the number of MDT/HDTs being used for 5th wheel RV hauling these days. On the other hand, all the hotshot drivers trying to sneak under the radar in 1 ton duallys have had a similar effect in some areas too.
Anecdotally, one of my personal trailers is 6" over legal width, and is a 14K GVWR...in 15 years of running that thing all over Arizona and California behind a F350 that looks very much like a commercial hauler wearing commercial tags (because Arizona considers EVERYTHING 1 ton or larger to be a commercial vehicle for license plates) I've never once even been stopped with it much less hassled or ticketed.