What are the advantages of an electron-electron collider vs. An elec./positron one? Has one ever been built? The current LHC is a proton-proton collider (instead of proton-antiproton) for reasons I have read about elsewhere.
Why has there never been an electron-electron collider?
Also, I am reading now about a newer, bigger electron-positron collider to replace the old LEP collider at CERN. (The thing that made way for the LHC).
But I read that the new lepton collider (if it ever gets built) will still be an electr/positr. collider, rather than an electr-electr.  collider in the style of the LHC,......
Why? 
 A: 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)
A: 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.
A: On the following subquestion in the body of your post - "Why has there never been an electron-electron collider?" - I would answer, because from the theoretical base of Standard Model, there is nothing new to expect.
In my post ( Should experiments of high energy electron collisions be done as an experimental test of the Standard Model? ) I have also asked about electron-electron collisions. But, the focus of this question was about electron-electron collisions as a clear test for SM. If energies of colliding electrons are enough, we should see electron-positron pair generation. This is an important positive control for SM which should be done, in spite of any theoretical confidence of such pair generation.
Why this test is important ? Just think what will be if no pair generation will be detected, not any positively charged particle. It will be SM crash, so this test is actually a crash test for SM.
