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Using an online distance calculator, the longest I can find are essentially tied: Madrid, Spain to Wellington, New Zealand, is a crow-flies distance of 12,388 miles.

Taipei, Taiwan to Asunción, Paraguay is a just barely longer 12,390 miles. However, you could probably shift the GPS coordinates within the city limits of any of those locations and make the distances match exactly. The circumference of the Earth around the eqautor is 24,902 miles, and around the poles 24,860 miles. Doubling the above distances yields 24,776 and 24,780 miles respectively, both just off the maxiumum theoretical limit. The pairs of cities are on almost exactly opposite points around the Earth. Other close contenders:

Jakarta<--->Bogota: 12,311

Quito<--->Singapore: 12,269

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