How can one tell, after a power failure, if the contents of a freezer warmed above freezing? Knowing this, and knowing the actual temperature in the freezer, will help decide whether food can be kept, or must be discarded.
The University of Ohio extension publishes a clever tip:
Place two or three ice cubes in a plastic freezer bag and seal. Keep this in the freezer at all times. In an upright freezer, you can have a test bag on each shelf. If there is a power outage you will know if the interior temperature was above 32°F if the cubes melt. If the cubes are melted, quickly determine the temperature of the water in the bag and you will know the temperature inside the freezer.
Of course, they also recommend having a proper freezer thermometer.
All these rely on the same trick - freeze something in a configuration it cannot hold when melted. If you come back to it still in that configuration, it never melted. If you come back to it solid, but in a different configuration, it melted and refroze. My variant is a plastic water bottle half full of water. Freeze it lying on its side, then stand it up. The advantage of this is that a water bottle frozen in this way is a useful thing to own on a hot day - fill it up with water and you will have icy cold water to sip from on a hike, for example.
If you wanted to get super smart you could mess around with freezing other solutions (salt water etc) so that you could tell (by which ones melted and reconfigured) just how warm or cool the freezer got to.
Before I went away for a month recently, I half filled a small paper cup with water, froze it, then put a penny above the ice. I figured if the penny wasn't still on top when I got back the freezer had lost power for some time.
In addition to using any of the number of frozen water-based indicators others have mentioned, you could buy what is called a hi-low memory thermometer, such as this one by Farmtek, and attach it inside the freezer. Most of these types of thermometers are made for outdoor use, so they are not likely to be highly accurate, but they should be sufficient for your purposes.
Find a geek to build you a Raspberry Pi or Arduino device with a temperature sensor which automatically logs daily temperature readings every 10 minutes and emails you if it drops below a certain temperature.
...I'll show myself out.