OSL-LHR-AMS/AMS-LHR-NYC/NYC-WAS/WAS-MIA/MIA-DFW-SLC/SLC-DFW-SFO/SFO-DFW-GUA/GUA-MIA-Caribbean/Caribbean-MIA-GIG/GIG-JFK-HKG-MNL/MNL-HKG/HKG-SIN/SIN-HEL-OSL

A couple issues:

1. You will have to pay for an extra stopover in NA.

2. 23 segments is 3 segments too many.

Solution either drop NYC-WAS and purchase separately (puddle jumper AE service) or take the train. Drop SA trip to GIG as part of ONE and reduced DONE4 to DONE3. Fly to GIG in conjunction with your NYC stopover to reduce NA stopovers; upon return take separately purchased air or train to WAS. This will bring your total segments down to 20 and reduce the number of NA stopovers to 6. Savings: about $1500 USD which should cover the trip to GIG from JFK.

OSL-LHR-AMS/AMS-LHR-JFK/<insert separately purchased GIG trip here>/WAS-MIA/MIA-DFW-SLC/SLC-DFW-SFO/SFO-DFW-GUA/GUA-MIA-Caribbean/Caribbean-JFK-HKG-MNL/MNL-HKG/HKG-SIN/SIN-HEL-OSL

Booked separately: JFK-GIG roundtrip options: in Y class and upgrade with miles (if you have AAdvantage miles). Options:
1. Purcgase a discounted Y ticket for $981 and upgrade 25000 AAdvantage miles and $250 co-pay each way. Total $1481 (if you have AAdvantage miles)
2. Or pay about $1800 for a B fare (that's a guess - you iwll need to call AA for the cost and availability of B fare) and upgrade with 20000 AAdvantage miles and no co-pay (if you have AAdvantage miles)
3. Or pay about $6000 for a J ticket.
4. Or fly coach class for $981.
5. Or save GIG trip for another time.