It's not all that competitive but you might want to consider an OW Circle Pacific fare. You'd be limited to N.Am. - S. Am. - SWP - N. Am. (If originating from N. America, it's either Asia or Latin America but apparently not both).

A route you could use would be YYZ-ORD-MIA/DFW-SJO-SCL-EZE-AKL-SYD (and whatever stops you can fit into SWP, SYD-MEL-PER) -LAX-YYZ. You're allowed 4 free stops with extra stops costing more. Not sure how competitive the fare is with a 4 continent OWE (I think the OWE does not permit crossing the same ocean twice so you'd have to do a 5 continent fare N. Am. - S. Am. - SWP - Asia - Europe - N. Am).

Doubt you can extend the fare even by one day.