One danger comes from multiplying the number of points of failure. Connections are more likely to fail than wires, each power strip may have multiple internal connections. So you are increasing the failure rate with each daisy in the chain more than just adding an extension cord.
Another danger is the temptation to overload the line because all the outlets are available. You start out just using a few them without undue load, but theres the free outlet and what could it hurt to plug in one more device? Eventually it looks like the picture in the fire safety handbook with a big red X over it.
Another problem, not exactly a danger, is that you increase your chances of accidently switching off or disconnecting a lot of devices at once.
Even electricians may stick to simple, problem free rules, like this one:
Simple rule for when to use copper wiring and when to use aluminum wiring: