What happens in electron-electron collisions? What are the results of high energy electron electron collisions?  Are other particles created?
 A: As far as I know, nobody has ever done this, at least not at what we currently consider high energy. (Electron-electron collisions happen at low energy all the time, of course.) I doubt that anything interesting would happen, primarily because electrons are mutually repulsive, and they have a low mass. That means two colliding electrons would just bounce away from each other. They would produce some photons from bremsstrahlung radiation, but there wouldn't be any nontrivial interaction.
Even if we did give the electrons high enough energies to get very close and possibly have some sort of interesting interaction - and we're probably talking at least tens or hundreds of TeV here - the possible products are constrained pretty well by conservation of electron lepton number and charge. In particular, the reaction products would have to have total charge -2 and electron lepton number +2. The only configuration of known particles that satisfies that condition is two electrons, and if there were an unknown particle that could change that, it would have to be very strange indeed. Other than that, some number of particle-antiparticle pairs could always be produced, but we get that already at higher energies from $e^+e^-$ and hadron collisions, so there's no particular motivation to build a machine that can produce electron-electron collisions.
A: I know my answer is 5 years too late, but I think it needs to be done. There were electron electron collision experiments. In fact, they were the first collider experiments that have been conducted.
So, electron electron collisions are also called Moller scattering which is well described quantitatively by the feynman rules for Quantum Electro Dynamics.
First, there is VEP-1 collider in Novosibirsk, Russia. It is a collider with an energy of $2*160 MeV$. It reached a luminosity of $4*10^{28} cm^{-2}s^{-1}$. The experimental studies on scattering of electrons were conducted and the following reactions were observed
$e^{-} e^{-} \rightarrow e^{-} e^{-} \gamma$ 
$e^{-} e^{-} \rightarrow e^{-} e^{-} 2\gamma$ (Double bremstrahlung)
So, the first reaction can be seen as a pair annihilation producing a photon. But, most probably it was from bremsstrahlung of either the initial or final states.
Secondly, there is also a collider in the US which is the Princeton-Stanford Experiment Collider. It was the most powerful electron accelerator during it's time. It is a collider with an energy of $2*500 MeV$. It reached a luminosity of $2*10^{28} cm^{-2}s^{-1}$. It has conducted several tests for QED.
Also, if you would like to read about these colliders here is the link: https://arxiv.org/abs/1307.3116 
A: Let's try just considering what could happen based on conservation laws. The two electrons have a charge of -2e, so the end product must as well. Lepton number conservation is required also, and we have $L_e=2$ here. At this level, it looks difficult to produce additional particles which satisfy just these two conservation laws. If you work in QED the only vertex is the photon one, and so if the energy of the two incoming electrons is large enough you might be able to produce something like
$$e^- +e^-\rightarrow e^-+e^-+(e^++e^-)$$
You can see the term in parenthesis are lepton-number and charge neutral. In principle you could do this with $\mu^++\mu^-$ or $\tau^++\tau^-$ as well.
