LOT has round-trip flights from Chicago (ORD) to Budapest (BUD) for $500-$586, NONSTOP. October to March. Departures on Tuesday and Saturday.
Note: Some small OTAs on SkyScanner have this for $500. If you plan on booking there, beware of any site that charges a fee to cancel a ticket within the 24-hour “free” period. Use unknown booking sites at your own risk.
Note: TAP Air Portugal seems to have flights on several of these dates for $479 with a connection, but you’ll be in transit for 6-hours longer than the Nonstop flights on LOT.