What are the advantages of an electron-electron collider vs. An elec./positron one? Has one ever been built?
There was VEP-1 electron-electron collider in 1960s, see arXiv:1307.3116.
The advantage of electron-electron colliders is that one only needs to deal with electrons, which is easier than dealing with positrons. The advantage of electron-positron colliders is that one can use the same magnets for electron and positron beams going in opposite directions
There was also Princeton-Stanford electron-electron collider (see the same link)
As akhmeteli says, there have been electron electron colliders. Note the date (1968) and the reason for making the collider:
The overall goal of the project was to check the limits of applicability of quantum electrodynamics at small distances by studying the angular distribution of elastic (Moeller) scattering of electrons by electrons.
In 1968 the quark model was still a revolutionary proposal for elementary particle interactions, and electron electron scattering was not useful for accumulating data. No matter how high the energy of the beams, the creation of new particles would be inefficient in energy to create particle pairs . Electron positron was much more efficient, also because of geometric considerations for the accelerators.
The situation is different with proton proton scattering at high energies. The protons are complex particles composed of quarks and gluons, so the deep inelastic scatterinq of protons on protons happens really by quarks and gluons scattering off quarks and gluons.
Electron positron colliders are good for detailed and accurate measurements of the interaction because of the simple vertices entering the Feynman diagrams.
Proton antproton and proton proton accelerators are good for going to higher energies and studying new energy regions, but not in detail. At high energy proton proton crossections are as large as proton antiproton.
The SPS, proton antiproton at CERN, was a discovery machine, looking for frontier data on particle physics and found it, the W and Z. LEP was built to check the standard model with accuracy, which it did. LHC was built on the same tunnel as a discovery machine, to find the Higgs, which it did, and expand the high energy data base checking the standard model. The proposed ILC will be built to get accuracy for the Higgs and test further the standard model, looking for deviations and new physics.