Is it possible to visit all six non-Antarctica continents, just flying, using only Low Cost Carriers?
Turns out, it is now possible, thanks to a post I found on Wikivoyage.
A sample itinerary, that they provide, beginning in London:
London to Barcelona on Vueling
Barcelona to Casablanca on Vueling
Casablanca to Istanbul on Air Arabia Maroc
Istanbul to Dubai on Flydubai
Dubai to Kathmandu on Flydubai
Kathmandu to Kuala Lumpur on Air Asia X
Kuala Lumpur to Melbourne on Air Asia X
Melbourne to Honolulu on Jetstar
Honolulu to Vancouver on Westjet
Vancouver to Cancun on Westjet
Cancun to Mexico City on Interjet
Mexico City to Bogota on Interjet
Bogota to Fort Lauderdale on Spirit Airways
Fort Lauderdale to Oslo on Norwegian Air Shuttle
Oslo to London on Norwegian Air Shuttle
Of course there are complications in doing so, as RTW tickets tend to imply a long trip, and LCCs often re-do their routes and cancel/change itineraries.
Recently, a Dutch guy reached it to the national newspapers, with is hobby of searching low cost flight tickets.
Apparently, he manages to get a lot of discounts, and collect frequent flyer passes with many airliners. For example:
I found a ticket from Paris for only 7 euros 50 It was a mistake of the airline, but I was going for a song in Asia.
He used this approach not only for this trip, but for trips all around the world, and to all continents.
Downside of his approach: he spends a lot of time looking for flights, and needs many transits on his travel. Furthermore, he books most flights more than a year in advance. However, it is fairly cheap for him (considering he is a student).