What EE/EM research can be done at home any more ?

[Psi]

Maybe invent some new silicon sandwich device? I saw a few videos of some teenage guy making transistors and IC's on silicon sheets, in his house.

Jerry Ellsworth made some transistors at home as well.

[mikeselectricstuff]

Without doubt, far more than ever before - dirt-cheap test equipment & tools, cheap manufacturing services from China, 3d printing, surplus high-end gear from ebay, massive computing power, access to huge amounts of information online. You can learn pretty much any skill you need from Youtube tutorials.
The only thing that's much harder now is finding something nobody's done before.

[Psi]

You can even mess around with superconductors at home.
Need to buy some LN2, which is not hard (just don't walk in there wearing a hoody and holding a 2L coke bottle)  .

And buy a puck of YBCO superconductor, which is a little spendy but well within the budget of a home researcher.  Under $500.   (or follow what Nilered did and make one if you really really want to)