Is it Safe to Plug an Extension Cord Into a Power Strip?

To be honest, I do it all the time. However, there are some things to keep in mind:

• Never exceed the maximum power, current and voltage.
  • Voltage will normally not be a problem since this is (more or less) stable in a country. When you go abroad, check if they can be used.
  • current is most important, sum up all devices you connect on that extension cord AND all devices on extension connected to the the first extension cord. Take some margin, especially motors sometimes can use a lot of 'starting' current.
• Don't forget when you switch on an extension cord (either by plugging it in, or by using a switch if there is on), ALL devices will be powered on at the same time, using possibly some additional extra current.
• When you go towards the maximum current supported by the extension cord or in a warm environment or radiation (e.g. from the sun), make sure the cable is fully unrolled (meaning not bundled), since warmth cannot spread (updated according comment of FerryBig).
• Apart from heating cables or connectors, a primary safety concern with an extension plugged into an extension of any type is that overall resistivity can become so large that even a full short at the furthest end of the cable may not trip a breaker. Another potential problem is that if the outlet is equipped with an earth-leakage circuit breaker safety device, that may also trip "on it's own" as the current in neutral wire is diminished due to excessive resistivity in the circuit. (addition from Kurja).

If you only plug low current devices into the extension cord there should be no problem. In any case, you need to keep the total current requirements of everthing plugged into the surge protector and extension cord below the rating of the surge protector. You also must consider the current requirement of anything else on the same circuit as the outlet the surge protector is pluggged into.

Manufacturers warn against plugging extensions or other power bars into a surge protector in case someone plugs a 15 amp heater into the last of three power bars.

I'll add some information to what Michael and Peter wrote in their good answers.

You are right about lots of warnings being written everywhere on the subject. This is some sort of scarecrow tactic from the manufacturers (and rightly so, IMO).

Manufacturers must design their product and the included instructions around the (possibly dumbest) average consumer. Average people, sadly, have very little understanding of how electricity works: they confuse power, voltage and current all the time, let alone knowing what Ohm's law or Joule's law are!

So manufacturers don't take any chances. They tell you the simplest possible rule to use their products with safety: "One extension cord (adapter plug, or whatever) to one socket, no multiple adapter/extension things joined together". Stop. Simple and hopefully foolproof.

Of course, extension cords, plugs, sockets and similar "electrical connection things" follow Ohm's law and Joule's law. This latter is particularly important to calculate heating. If you know what you are doing and know the characteristics of your loads, you can mix and match almost anything by providing generous safety margins.

I have been running various multiple extension cords setups for ages, without the slightest problem. "Normal" people could watch my setups and cower in terror (if they are sane), or they could do the same and burn their house down, because they can't tell the electrical difference between a 4W power pack and a 1500W toaster! :-D