EVA Air and Starlux Airlines have round-trip flights from Los Angeles (LAX) to Taipei (TPE) for $896-$941, NONSTOP. Flights August, September, October, November, and January. No Weekends. Departures on Monday, Tuesday, Wednesday, and Thursday. Dates over Thanksgiving are available.
Some small OTAs on Skyscanner and Google have this for cheaper than the airline. 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.