You can do this using a Oneworld Explorer as a four continent RTW, with a basic routing such as:

SCL-AKL-HKG-DPS-HKG-FRA-SCL

You would be able to add four segments in South America, Europe and Oceania and two additional segements in Asia if you were so inclined.

At the moment, the cheapest place to start a RTW is Egypt, which, if you live in Chile (and I am jealous of you if you do!), would be a long and presumably expensive trip. The problem with starting in Chile is that fares are the same as starting in the USA, which is one of, if not the most expensive, place to start a RTW.

You might also consider the Oneworld Global Explorer, which is a mileage based ticket and may be cheaper for the itinerary you are proposing.