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Short answer, it's actually not.

The actual danger is overloading a socket. This is significantly easier with a chain of extensions.

Resistance on the individual plugs shouldn't be an actual issue. The only way it might come into play is if you are using a switch mode power supply at the end. This would adjust its current draw to account the reduced voltage. This would increase the current draw at the plug.

E.g. a PSU trying to draw 1000W. At 120V this would be 8.33A. If you drop 30V, then the supply would draw 11.1A. The intermediate plugs would dissipate 333W. You would be drawing 1333W from the wall.

In countries with low AC voltages, and a lack of fused plugs, this could overload a socket and cause a fire.

It's also worth noting that long extensions have the same issue though it's less. They also have a far lower rating when coiled than when uncoiled. E.g. I have a 13A extension that can only provide 4A when fully cooled.

Other commenters have pointed out the problems with overloading of connectors and reduced efficiency because of the added resistance but there is another really important reason not to chain power strips: circuit breakers work best against short circuits when the resistance between the breaker and the short is fairly low (for instance less than 0.5Ω) so that the current will quickly go over the rated current of the breaker. If the resistance is a lot higher because you have too many extensions between the breaker and the fault, the time the breaker needs to react will go up. Counterintuitively this usually means more energy will be turned to heat by the fault.

In extreme cases this can mean the difference between a broken power strip that you can just throw out and a burned down house.

The longer the run with more connections, the greater the voltage drop, the more stress on anything involved.

That means more power usage and higher likelihood of a failure vs directly plugging the same device into the outlet. So it is better to use an extension cord than chained power strips.

It is especially dangerous to have chained power strips that you load up due to fire risk that isn't necessarily mitigated by the breaker.

It's complicated. If everything is rated properly, it's not really an issue, the problem is you can hook a 15A power strip to a 10A or less extension cord that then goes into the wall. The strip will let you plug in 15A worth of equipment but its upstream can only handle 10.

But if you have the right sized extension cords and power strip for your load it's fine. It's just, the average person doesn't really understand the nuances.

https://youtu.be/K_q-xnYRugQ

Are you talking about extension cords with multiple outlets at the end?

The wire inside the cord could be of unknown quality - typically these can be either really thin copper wire, or aluminum/steel which are not ideal. The cord can get hot under heavy use, and in a worst case scenario, melt and start a fire.

Too many extension cords chained together can cause the mains voltage available at the end of the cord chain to drop, due to the resistance of the wires adding up.

Each individual plug and socket are potential failure points, and like the cord, the materials used for the socket's contact pins may be less than ideal. The less of these loose points you have chained in any kind of electrical system, the better. The resistance at these points may also cause the contact pins to get hot under heavy use, causing the plug to loosen over time.

Typically the fuse breaker should cut out and protect the wires before anything dangerous happens - but IMO better safe than sorry.

I might have done a poor job of explaining this... hopefully someone else provides a clearer description

Power dividers are RF components, they divide power (and in reverse they're power combiners), if you chain 10 power dividers you get around 30dB attenuation. If you want to compensate for that make sure your system can handle more power.

For extension cords with multiple sockets read the other replies.

Each connector has a little bit of resistance and thus increases heat. This is compounded by the voltage loss, so each one gets a little bit worst.