What is the lowest possible theoretical temperature that nuclear fusion can occur at? I am not talking about the pseudo-science of so called cold fusion  I am interested in what temperature you can get away with to produce fusion reaction. I was thinking in terms of micro-fusion or at least at levels below the huge tokamaks being produced because obviously small scale devices couldn't handle millions of degrees of temperature. Or correct me if I am wrong please as I am no expert just a curious student!!
 A: The required conditions for fusion depend on density as well as temperature.
In extremely dense conditions it is possible to initiate fusion via quantum tunneling. In principle, pyconuclear reactions can occur at zero temperature in condensed matter because the ions sit in a crystalline lattice and have a quantum mechanical zero-point energy. If the material is dense enough then this zero-point energy can be sufficient to tunnel through the Coulomb barrier and cause fusion.
In astrophysics, pyconuclear reactions may well be the starting point of a fraction of type Ia supernovae or cause instabilities in massive white dwarfs before they reach the ideal Chandrasekhar mass (see Boshkayev et al. 2012). The rates of these reactions have been calculated since the late 1950s (e.g. Cameron 1959; Salpeter 1961), but are highly uncertain. At the moment, I believe the situation is that pyconuclear reactions could be important in low-mass white dwarfs that accrete a lot of helium; might play a role in carbon white dwarfs; but probably aren't important in elements that are heavier than this because of the larger Coulomb barrier. 
I realise your question was more focused at fusion that can be initiated in a terrestrial setting, but to my surprise I find some mentions of the possibilities of "cold" pyconuclear fusion (not at all the same as "cold fusion"!) in a number of respectable places. For example see p.14/15 of this Lawrence Livermore report on the "National Ignition Facility", or this US patent. 
A: ITER is aiming for 150.000.000K. Please note that this temperature of the plasma, i.e. average kinetic energy of the ions is in electron volts

For example, a typical magnetic confinement fusion plasma is 15 keV, or 170 megakelvins .

15 KeV is enough to assure that the plasma does not neutralize itself and the bare nuclei have a high enough statistical and quantum mechanical probability of interaction/fusion by overcoming the coulomb repulsion of two positive nuclei.


The electrostatic force between the positively charged nuclei is repulsive, but when the separation is small enough, the attractive nuclear force is stronger. Therefore the prerequisite for fusion is that the nuclei have enough kinetic energy that they can approach each other despite the electrostatic repulsion.

The temperatures necessary for the various fusion reactions:


The fusion reaction rate increases rapidly with temperature until it maximizes and then gradually drops off. The DT rate peaks at a lower temperature (about 70 keV, or 800 million kelvin) and at a higher value than other reactions commonly considered for fusion energy.

This is in a nutshell the status for thermodynamic/plasma fusion.
