Rats. CX site wasn't working properly last night so I couldn't double-check, but the sticky thread listed CX as having "some flights" HKG-DXB in First.

Okay, how about this:

DEN-LAX-SYD-BNE-ADL-PER-SYD-HKG-NRT-SIN-BKK-HKG-LHR-DXB-AMM-DME-LHR-YVR-JFK-MIA-DEN

20 legs, 48,757 miles and valid according to Mileage Monkey and the oneworld tool.

I'm not wild about stopping in AMM, but the only other way out of DXB seems to be returning to LHR, which in turn would mean no DME because we'd exceed the limit on flights from LHR to certain destinations.

It also puts us in Business on Royal Jordanian for two legs (DXB-AMM-DME), but I guess that's supposed to be a good product?