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I think a lot of people are mostly on the money here. It's to do with resistance. Now, I'm not a qualified electrician, but I'm an amateur radio license holder and a lot of what you learn for that is applicable here.
The main problem as many have said is resistance. This comes about from both the length of the conductors but also from every plug/socket connection adds resistance. Also in the case of the non extension socket multipliers, as you add more the weight bearing down would also likely start to make the connections less secure causing more resistance and possibly adding to the problem through arcing.
Now the resistance alone on small loads likely wouldn't be a huge problem. But if you had a large enough load (specifically at the end of the stacked connectors/extensions), or a fault that caused a larger than expected load the current would cause the resistance to generate heat.
There's a lot of ifs and maybes involved, but really why do it? There's really no real world situation to need to have a dangerous amount of extensions like this though.
For larger loads here in the UK there's some very specific other concerns when dealing with ring mains. But really you'd need to do really weird/unusual things for that to become a problem.
On the contrary, practically every single person in this thread that mentions resistance is talking precisely backwards. Higher resistance will not cause overheating. Except for certain motor loads, it will cause malfunction due to voltage drop. The actual fire hazard is the increased chance of faulty cable, faulty connection, and, inevitably, overloading.
Resistance alone doesn't cause heat. Drawing current through resistance causes heat at the point (or points) of resistance. Which is why I clarified that it's not likely so much a problem on small loads.
This is why resisters come in different physical sizes. Because they have differing abilities to dissipate power as heat.
A good example is of dummy loads in radio use. Which needs to dissipate the power output of a radio. That can be anything from milliwatts to a kilowatt. Up to probably 50w they will have a basic heatsink. I've seen huge drums filled with oil as 50ohm resisters to handle up to a kilowatt of dissipation.
Yeah, but shunts are "added" in parallel, daisy chaining cords adds them in series.