Do expedition bikes offer advantages over standard touring bikes? I have seen a few bikes marketed as "Expedition Bikes". They seem to be designed for the purpose of touring off-the-beaten-track. Typically they have 26 inch wheels and either butterfly or flat bars, though I have seen a couple with drops. Some even have internal hub gears. The one defining characteristic seems to be the 26 inch wheels though. 

What is the advantage, specifically, of the 26 inch wheels over 700c when on an 'Expedition' type tour? And more in general, are there other advantages to buying (or building) an Expedition Bike rather than just modifying an existing road tourer?
You should find that on those bikes everything is a bit stronger and heavier than a standard touring bike. Not only are they expected to carry more weight, they're designed to be ridden into places where failures are more difficult to recover from. As well, because they're designed to be ridden off road they'll usually have a lower top tube for better stand-over height, and there are fewer compromises in building a frame like that with smaller wheels.

Smaller wheels are stronger than larger ones, mostly laterally but also weight-for-weight vertically (shorter circumference means more material per unit length). This is most obvious with the 306 and 406 wheels used on trikes, but applies also to larger wheels except that the differences are smaller. Larger wheels mean the axle is higher, but also the top of the fork crown and so on, so you're adding the change in diameter to the frame height, not the change in radius.

Decent 26" tyres are more available in small towns. With a 700c or 27" wheel you may find that you have a choice of flimsy, skinny tyres and nothing else. (thanks Batman). In a pinch you can tour on cheap MTB tyres more easily than cheap 700c tyres, because a cheap, skinny tyre is going to puncture more easily and not deal with touring loads as well. I've seen a lot of bike shops in small towns in the first world that have BMX tyres, 26" off-road and city tyres, and 700c "race" tyres, all cheap unbranded tyres. It's not a choice between an expensive good brand and a cheap one, it's a choice between two widths or styles of the cheap one. If you have a choice. Or wait a week or more for mail order, because your order goes by air to a major city, then road to a minor city, then bus freight to where you are. Ideally you'd always carry a spare tyre, but in practice that just means you're waiting to replace the spare rather than the currently wrecked tyre.

Another factor is ease of building a frame that will accept fat tyres with the wheel size. Once you add a 2.5" tyre to your 700c wheel the frame gets quite large (29er MTB frames), which makes designing for low stand-over height harder and all the stresses on the frame get larger because the tubes are longer making for longer levers. The differences are small, but they add up - a slightly longer lever needs a slightly stronger tube, so now you have a longer tube that's also heavier per unit length, and overall the 10% extra length means a 25% heavier tube.

You are also more likely to find disk brakes on expedition bikes, because again they're more robust. Not necessarily the caliper and rotor, but the wheel as a whole. If you get a buckled wheel, with disk brakes the critical question is "does the buckle clear the frame" (ie, does the wheel go round), but with rim brakes it's "can I adjust the brakes to clear the rim" (that ignores "do the brakes still work like that").

The reason you see Rohloff hub gears on expedition bikes is the same. Sure, they add cost, but they are reliable. Not only do you lose the low-hanging, fragile rear derailleur, the straight chain line and high chain-ground clearance means chains last longer. The hubs go 5000km between oil changes and 150,000km or more before needing replacement. As a bonus you get a stronger wheel from wider spaced, symmetrical flanges (that are also taller). Again, it might be 1kg heavier, but when you're loading 50kg or more of stuff on the bike, do you really care?

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The motivation for a lot of this is that it's all very well saying "I'll just get a lift to the next town", but if you're riding the Birdsville Track in the off season (recommended to avoid getting run over by tourists) you might wait several days for the next car to arrive. And they might not be able to carry your bike as well as you. So not only do you need enough food for 500km, you need water for at least a week (2-3 days between water supplies, plus a reserve). Then you have to get a lift back to your bike with parts, fix it, and ride on. Much better to pay a bit extra and get a bike that doesn't break in the first place...