Thanks for the replies. As I understand so far, the options are:

1) Start in Bangladesh - not likely to happen; I would really rather go ex-NA, ex-Europe or ex-Tokyo
2) As was pointed out already, LHR-DXB-LHR is not possible starting in Europe
3) As was also pointed out, ANC is not possible on OW, but YVR and PTY are alternatives
4) The South America alternatives are interesting, but I have flown those routes so many times that I would rather go on other routes; also, I am trying to flying non-AA metal (I think that F and J on QF, JL and CX are better)
5) SIN might be a better base for mileage purposes in Asia than HKG (although I would miss out on the CX cabins)
6) It seems that PER cannot be in any OW itinerary because of the restriction on the one PER leg, unless I purchase more segments in Australia/NZ


Taking all this input, I came up with:

SJU-ORD-LAX-LHR-DXB-LHR-IST-LHR-MEL-AKL-BNE-PER-SIN-NRT-DPS-NRT-HKG-JFK-SJU-LAX-ORD

However, this only gives 64445 miles.