What is Hagedorn Temperature? I would like to get an understanding of the value of the Hagedorn temperature and the units this temperature can be given in. Is the Hagedorn temperature the maximum temperature, just as $0\ K$ is the lowest temperature? What happens to matter when the Hagedorn temperature is reached?
 A: The Hagedorn temperature is a maximum temperature in the same sense that the boiling point of water is a maximum temperature: namely, it's the maximum temperature for a particular "phase" of matter, and going beyond that temperature requires a "phase transition". In the case of water, this phase transition is evaporation (aka boiling); once the water has evaporated, it can begin to reach higher temperatures. In the case of hadronic matter, the phase transition is the quark-hadron phase transition; above the Hagedorn temperature, quark matter takes the form of a "quark-gluon plasma".
A: Hagedorn Temperature is the temperature at which the strong force gluing quarks in hadrons is exceeded by quarks' vibration energy, thus making ordinary matter disintegrate into quark matter. After matter conversion into quark plasma, this plasma can be heated further until Planck temperature is reached. So Hagedorn Temperature is not highest possible particle ensemble temperature, but just the temperature of phase transition $T_{\text{matter} \to \text{quark plasma}} $. Maximum possible temperature vs. Hagedorn Temperature ratio is  $$\frac {T_{\text{Planck}}}{T_{\text{Hagedorn,Quarks}}} \approx 10^{20}$$
In String theory, a separate Hagedorn Temperature can be defined, this time not for hadrons, but for strings. In effect this means that quark plasma is disintegrated further into "String Plasma", i.e. interaction forces between Strings are exceeded by String energy, thus breaking matter further into a soup of strings. This string phase transition temperature is very near Planck scale:
$$ \frac {T_{\text{Planck}}}{T_{\text{Hagedorn,Strings}}} \approx 100 $$
So it's not likely that this can be tested in laboratories in the near future.
