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There is actually no mystery to this at all - and it is something almost unique to their regular ham/cheeseburger. The main reason that food decomposes is bacterial growth. By a quirk of their design, McDonald's ham/cheeseburgers are a poor evironment for this to happen. The secret is the amount of water and the shape.
1. A standard McDonalds patty weighs 45g (or 1/10th of a lb, basically). That 45g contains a relatively high amount of fat - something like 7g from what I can gather. Like all McDonalds food, it's fat instead of water that gives the juiciness. Water is actually undesireable in batch cooking products, so the water content has been lowered.
2. That 45g patty is presented as a thin, wide shape. This makes it very easy for what moisture you do have to evaporate... even inside the bun.
3. The high fat content of the burger means the condiments won't penetrate the meat, keeping it dry. Basically it turns into McDonald's jerky!
What these videos don't show is that the bun will have rotted at least some where the condiments inside are placed. The shape and weight of the bun allows it to dry out (like the patty), but it will certainly decompose a little inside... it's simply that the bun will dry out completely before rot can fully take hold. Fat is what keeps the bun, patty and fries looking anywhere presentable - but it's the simple absence of water that stops it rotting.
Any other product from McD's would rot - it's simply that this product is a 'perfect storm' of conditions to prevent it.