EVA Air has round-trip flights from New York (JFK) to Taipei (TPE) for $659, NONSTOP. Flights September, October, November, February, and March. No Weekends. Departures on Monday, Tuesday, Wednesday, and Thursday.
Note: Vayama has this for $627. If you plan on booking there, be aware that they do not honor the 24-hour “free cancellation” period. They will charge a $100 cancellation/change fee regardless of when you cancel.