>crossing the antipode of your starting point at some point in your journey
In that case I pretty much have to start in Hawaii, since the rest of North America is opposite nothing but airportless rocks in the Southern Indian Ocean. Also helps that my longest overwater leg will be the first, so I can prep for it the best. Most economical is probably just going to be a 172 with the back seats being long range ferry tanks
>Hawaii to California
>Cross the US to Florida
>Cross the Caribbean to South America
>Brazil to Africa or St. Helena
>Reach Botswana (antipode)
>Up the East Coast of Africa
>Along the Southern cost of Asia
>Island hop back to Hawaii
Obviously will be a pain in the ass and you can bet your dick I'll have my plane loaded with enough survival gear and signaling equipment to maximize my odds of making it if I go down in the ocean
>guesstimate milage: 23,000nm or so
>estimate flight time @100kts: 200 hours
>estimate wet cost @$75hr: $15,000
Realistically I'm doubling that cost to account misc flight costs. Between everything else I'm betting keeping the cost under a million would be pretty easy

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What's the least you would spend on a round the world trip?