I've been on 23s for so long that anything bigger is 'fat' to me. With the advent of disc brakes on road bikes, it's now more common to find 25s and even 28s on them, but at least for road applications, I don't see tires getting any fatter than that. Once you get to 32, you're looking at gravel-capable, and while that may be the whole point, there's a whole swath of cyclists who just don't do gravel - or if they do they have a dedicated bike for that and it's got knobbies.
A lot of fellow club members have tried fatter tires, 35-40mm, and while they say they're *almost* as fast as skinny tires, there's that "almost" qualifier. They can be softer-riding, though.