I've only just noticed this interesting thread. Putting in my pennyworth...
It is many years since I studied electrical theory so if anyone want to
correct anything I say please feel free :-)

1. The ability of a conductor to carry current is proportional to its cross
sectional area.
2. The resistance of a conductor varies with temperature (to a lesser or
greater degree depending upon the material) with some materials resistance
increases with temperature but with others it decreases.
3. Inductive effects can be a problem with long cables depending upon the
material used to make the conductor and any surrounding conductors which
'may' contribute a dampening effect. If a cable is left wound on a reel this
will have a larger inductive effect than if the cable is laid out straight.
A wound cable is also not recommended as any heat loss in the cable cannot
readily escape, so if a cable is heavily loaded this could cause it to heat
up and possibly melt.

In other words running an extension cable rated for 15 Amps at mains AC
voltage can safely and without problem run an appliance of up to 230 x 15 =
3450 watts (3.45 kW).

The maximum length of the extension cable is a function of the resistance of
the cable (should be negligable) and inductive resistance (again should be
negligable). Without exact figures for the parameters of the cable I suspect
these factors are unlikely to be an issue with modern copper cables. This
may have been an issue years ago with poor quality cables with impure copper
or an old extension cable that had been 'bent' so many times the copper
conductor was starting suffer from micro fractures?

I once did an experiment (outdoors) feeding an electrical kettle with a
cable rated at 2 amps. It was heavily overloaded but surprisingly the cable
just got hot and limp. The kettle eventually boiled too, but it took much
longer than normal, so there were significant losses in the cable. Pushing
it to the limit I added a second electrical kettle to the load. This time
the cable melted and failed with a nice big flash!

So while there is an element of truth in the original posters comments, I
very much doubt there would be any significant losses running an electrical
mower on a modern, good quality, extension lead that was within rating.

A slight tangent:
As power is given by voltage x current the same power can be delivered by
increasing one of these parameters and lowering the other. However, since
the maximum current that can be delivered through a cable depends on its
cross sectional area then using a higher current rather than higer voltage
is not cost effective as it uses more conductor to transmit a given power.
So high voltage and low current is prefered. That is why pylon wires are
such high voltage (used to be 143,000 volts as I remember?). However, with
higher voltage another factor comes into play - corona discharge. As pylon
wires are not covered in insualtion electrons at high voltage have a
tendency to jump off imperfections on the surface of the cable into
surrounding air molecules or water molecules. The effect is more pronounced
in foggy weather when this effect can be audible as a "zizzing" sound.

Norman Digger.