Your final choice would be to book a low cost carrier round the world flight that aren’t booked on a single ticket. For example, book a Jetstar flight from Auckland to Bangkok. Then book a Norwegian Air flight from Bangkok to Copenhagen. This will get you a pretty low cost RTW flight, but there are risks.
The big one is that these flights won’t be on the same ticket. So, if delays or cancellations affect your flight, that’s your risk. This could mean you get delayed in Bangkok for a few nights and miss your flight to Copenhagen. In this situations I recommend including a night or two in that city to reduce your risk. These delays do happen – a friend of mine was in Tokyo flying Jetstar to Melbourne. Her flight delay was three days due to engineering issues. This means more hotel expenses, which in same cases are an out of pocket expense to you, than you then have to claim back. So it could be a cheaper deal, but that’s because you accept the risk.
Maximising usefulness of low cost carrier round the world flights
Of course these tickets provide you with the most flexibility and the most risk. Basically book any airline, any time, just find the best deals. You don’t have to worry about airline alliances, really just focusing on what airlines travel to what destinations. Wikipedia is a great way to find out what airlines service an airport, for example, in this article about Auckland Airport. Then search Google Flights for flights between those two destinations.
Low cost carrier Round The World flights will often get you to destinations you never considered. One of my favourites is flying WOW Air to Iceland. Iceland is an amazing destination, and flying a low cost carrier is a great way of getting there. I do recommend paying more for upfront seats so you get more leg room, and a more relaxed boarding process.