Nearly everything a Kent remodel produces can be loaded. Kent's own rules draw a hard line around four categories: hazardous waste, asbestos, asbestos-containing material, and infectious waste may not go into any public or private container in the city, and that applies to a rented container as much as a curbside cart.
Beyond those, the usual exclusions hold. Liquids, paint, solvents, fuel, used oil, batteries, and pressurized cylinders all need separate handling. Tires and large appliances should be set aside and mentioned when you book, because they are normally billed by the piece rather than by weight.
Kent also has a packaging rule that catches people out. Loose material such as rock, brick, block, dirt, sand, and cement has to be properly contained rather than loaded loose. A container handles that requirement for you, which is part of why the city names a larger receptacle as the fix when a property outgrows its carts.
The city further requires every load to go to a site properly licensed under state law, so the destination is not a free choice. Anything you cannot place on that list is worth raising with your account manager first.