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[Virex]

I dont understand why finding out the flame spectre (which should be easy enough) is off-limits and complicated methods which involve millimetric measurements of very hot nails (and which at least at a glance seem rather error-prone) are not 

Because
A: the flame spectrum is dependent on the burned chemicals, largely temperature independent and
B: the blackbody radiation by a flame of this type is hard to detect, because of the blue ionised air. The color blue in blackbody radiation spectra is about 10000K.

Although you could measure it using an Infrared temperature reader, if you have one lying around with a scale that's high enough...

Things are about to get worse: gasses don't have blackbody radiation, instead they have spectral lines. Even in the infrared area the colour's completely bodged up because there's no complete spectrum. So an infrared thermomether is going to give you senseless readings.
It can be done, but you'd need to know the exact composition of the gas in the flame itself and then use statistical thermodynamics (bolzman distributions in this case) to calculate the temperature from the quotient of 2 spectral transitions. But that'd probably give you a headache and you need a spectrometer to boot.

[Cthulhu]

He's a junior in high school, I think you're going a bit farther than the teacher was expecting.

[qwertyuiopas]

Well, if you want a general range, find about 20 objects with known melting points that are roughly evenly distributed between 100 and 2000 degrees. (About 100 degrees diffrence between each)

Then see what ones will melt.

Alternatively, can the time to boil a set ammount of water be used?

[zchris13]

That's just the heat in the damn thing, and how much gas you're burning, and how effectively it transfers.

Not the temperature.

[Siquo]

Well, if you want a general range, find about 20 objects with known melting points that are roughly evenly distributed between 100 and 2000 degrees. (About 100 degrees diffrence between each)

This! This is the one!

A: It's easy to do, you just have to find some materials, but the 'evenly distributed' is not really necessary, just as long as most of them are around the point of temperature you guess the burner is.

B: You get to see things melt, burn, or sublimate! Chemical vapours galore! This is how HFS determines if they have bad breath.