I’m floatin’ on up.
Is vacuum actually the best insulator available? I don’t think it has to be. On the one hand, it does not convey any heat through convection, but it also is transparent, so any radiated heat will go through it. Might some crazy material (or electromagnetic field) be able to reflect enough radiated heat that it would prove to be a better insulator?
An Atoic Balloon?
Ranchoth aspires to be the Father of the North Korean Air Force!
Couldn’t you somehow refuel while still in flight? That is, could a second aircraft of some sort (another balloon or a small helicopter or something) be launched, shoot or lower a tether to the balloon, and then send over some more fuel? Surely there must be some way of positioning a helicopter above or beside a balloon in calm winds such that it won’t severely disrupt the balloon’s flight.
What you want to do is to have a layer of vacuum, to stop convection and conduction, with a layer or two of polished metal or other highly-reflective surface, to stop radiation. Which is, in fact, how a thermos is made.
The balloon could lower a rope to the ground for this I guess. Also I have seen (on TV) a skydiver purposly landing on a balloon, so no reason he can’t bring fuel
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So there is your upper limit of heat loss (well that would be lower limit).
I thought of that, but then I figured that if you’re tethered to the ground some people will argue that you are no longer “in flight”.
It depends on the goal. If you’re trying to set a record, it’s much more exciting to set an unfueled distance record. And I can’t think of any other good reason to do it - if you want a balloon for carrying useful payload, a helium balloon would be much more practical.
Nuclear? Good Gawd, I have no idea where to begin, as I have no idea what that would entail.
I just wanted to chime in, and let the dopers know that I have been involved in hot air ballooning my entire life, and could answer any questions you might have about existing balloon technology.
Of course, now that I type this, I realize that I know a bunch about conventional ballooning, but not a ton about around-the-world varieties.
Maybe it would be best to pat me on the head and send me on my way.
One other thing…
As the OP mentioned, the balloons that circumnavigated the globe were not purely hot-air powered. Effectively, they are almost purely helium powered, with hot air employed only to heat the helium.
I guess that goes without saying, but I thought I’d throw it out there…
Good suggestion, but I was after something more evocative of the Golden Age of lighter-than-air flight. Plus referencing the anti-radiation shielding, of course. You know, “Lead Hindenburg” or “Lead R-101” or something… Dang, the Muse has obviously popped out for a cigarette break. 
How about the Frying Macon ?
I suppose it’d get bigger if you add extras like shielding and stuff, but the core unit is at least in the right size/mass range to power a balloon.
How about the Dread Airship - you could then have a cover band of it called Dreadnaught which would cover its songs in a reggae style.
I’m sure it would look majestic flying over some Misty Mountains. And it could be useful for delivering supplies to New Orleans when the levee breaks.