An interesting question. IMO the answer is no/very little. I feel like vegetables are a fine accent to a meal for kids if they like them, but its of little to no use to try and force them on a kid. Kids burn through so many calories that veggies really arent even the best use of their tiny stomach space.
I fill mine up on fatty meats and eggs first, fruits and starches second, milk/cheese third, and some nutrition devoid rice products last. Somewhere after all that I'll ask em if they wanna carrot