What are some "Illegal" LEGO combinations?
I do not think it is up to us to determine what is considered an Illegal Technique - TLG (The LEGO Group) determines this for their commercial products. It depends on how much of a purist you are whether you apply these constraints in your own builds. 
That said, LEGO Designer Jamie Berard has made a presentation in the past that highlights most Illegal Techniques.

Some examples:


Studs-on-side bricks cannot have their side stud in the hole of a
technic brick
Bricks mounted SNOT style onto Ehrling/Headlight bricks cannot sit
directly above a normal stud (the minimal height of the LEGO
embossing makes that combination illegal)
Technic pins have to be fully inserted into technic holes, otherwise they stay in compressed state which causes stress on the element.
Plates cannot be inserted with their studs into technic bricks, unless it is a 1x1 plate (so only 1 hole is used) and there is no brick attached to the technic brick directly above the inserted plate.
As mentioned by you already, you cannot insert plates between studs (this used to be a valid technique 20+ years ago, but is no longer considered "legal" by LEGO).