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If you want a numerical value, then read no further - you won't find one.

Temperature can be regarded as a measurement of the density of heat energy in an object or region of interest. What this means is that the only limit to how high a temperature can go is how much heat energy can be accumulated, and how small a space it can be confined to. That limit, then, would be to have all of the mass-energy of the universe (remember, E=mc^2) converted into heat energy and contained to a single point of space. (This would be akin to the state of the universe pre-Big Bang.)

The mass of the universe is not known (though varying estimates are made regularly), so the total energy is not a calculable value.

Practical limits also exist, since heat tends to radiate, so that energy would constantly be escaping, thereby cooling our hypothetical container before we could reach the mathematical limit. The only way to avoid this would be to eliminate any space outside of our container, making the container the entire universe, again bringing us back to pre-Big Bang conditions.

As far as observable temperatures go, the cores of stars can reach temperatures of 100,000 Kelvin (180,000 Farenheit), which is a little on the toasty side.

Learn more about this author, Ernest Capraro.
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