A typical (cheap) extension reel (10 metres, 10 amps) has the following specifications:
Uncoiled 2400W (10 A) 240 V
Coiled 720 W (3 A) 240 V
The reel had 4 sockets and a 13 A fuse in the mains plug.
Measuring the lead resistances I found them to be approximately $0.3 \; \Omega$ and $0.4 \; \Omega$ so a total of $0.7 \Omega$.
Suppose that the maximum current $13$ A hen the power dissipated in the whole cable would be $13^2 \times 0.7 \approx 120$ W.
If the reel is tightly wound and assuming that the electrical insulation on the cables is also a good insulator of heat you have the potential for quite a rise in the temperature of the cable even if some of the $120$ watts is lost through the outer surfaces of the reel.
In the past I have certainly noticed a wound up reel getting hot (rather than just warm) so unless I am using small power devices have always at least partially unwound a reel.
A quick sum shows that the radius of the copper conductor has to be a rather low 0.4 mm to give a resistance of 0.35 $\Omega$ for a 10 m extension reel so I am undertaking a deconstruction job on the extension reel.
Even if the resistance of the copper cable is a factor of $10$ were and the power dissipated is of the order of $10$ watts this still has the possibility of raising the temperature of the reel to a dangerous, melting insulation, level.
The cable used had a copper conductor cross sectional area of 2.5 mm$^2$ and PVC insulation rated up to a maximum working temperature of 70 $^\circ$C.
The total calculated resistance of a 10 m reel is 0.13 $\Omega$ and so with 13 A passing through the reel the power dissipated as heat is 22 watts.
Still sufficient over a period of time to raise the temperature inside a tightly coiled reel to a level where the temperature rise will be quite high.