United's contract of carriage is actually quite explicit about this. Your mileage may vary as to anything they don't explicitly say you get for free, but, look under "carry on free baggage allowance" on page 22 of
http://www.united.com/web/format/pdf...f_Carriage.pdf and you'll see:
d) In addition to the Carry-on Free Baggage Allowance listed above, the following items do not count toward
the one Carry-on plus one personal item:
(i) An overcoat or wrap.
(ii) An umbrella.
(iii) A reasonable amount of reading material.
(iv) A pet carrier (fees apply) (the carrier must be small enough to fit underneath the seat without blocking
any person’s path to the main aisle of the aircraft, and it must be stowed properly before the forward
customer entry door to the aircraft is closed).
(v) A collapsible wheelchair.
(vi) A government approved child/infant restraint seat meeting Federal Motor Vehicle and FAA Approval
Standards.
(vii) A camera.
(viii)A diaper bag
(ix) A limited amount of Airport Duty Free items, merchandise purchased in the airport, or food. These
items must be stowed in the same manner as Carry-on Baggage.
(x) Assistive devices (a cane, one set of crutches, prescription medications and any medical devices needed
to administer the medications, a Portable Oxygen Concentrator (POC), etc.). These items must be stowed in
the same manner as Carry-on Baggage.
If you are carrying an extra bag that has blankets (not diapers), you will find that some gate agents will make you consolidate. Others will not.