Some combinations of pieces are really hard to get apart – really hard. Can some answerers provide a list of “dangerous combos“? For example, a 2×2 with a round 2×2 on bottom. Also, if possible, provide a good way to get them apart!
A 1x1 round plate on the bottom of a 1x1 square plate is pretty difficult to get apart. The Santa's Workshop set uses transparent 1x1 round plates stuck until 1x1 plates with a clip on top to make Christmas lights. I just left the whole thing assembled; I'll probably never bother taking that apart.
Also: if you stick some brand-new 4x4 plates together it can be very very hard to separate them. Lots of clutch, lots of rigidity, so even with 5 brick separators it is difficult work.
My son did some extensive research on this subject, which yielded this beauty:
That’s a light bulb cover stuck into a Duplo stud. While it may look harmless, I found it impossible to get sufficient hold of the lightbulb before levering it out with a very thin screwdriver.
I have a 3062b (round 1x1 open stud) stuck in the middle of a 6067 (boat mast section base 4x4) and I can't get it off. The round 1x1 fits over the post thingy in the center of the mast base and the side walls of the mast base prevent access to the grove at the bottom of the 1x1 round. The inner diameter of the mast base is small enough and tight enough that any antenna type pieces inserted through the bottom can't catch the lip at the top of the 1x1 round on the way through.
Also, part 99009 and Part 99010 is another one that is extremely hard to disassemble as highlight in this question. (The current brick separator will not work in this situation.)
Generally, early BIONICLE body bricks can encounter issues with this, due to their bulky, specially-designed nature. For instance, the Toa Mata body gearbox has a particular danger area inside it. If one places the #2 axle connector hub horizontally inside the neck area and sticks a 2-long axle in either side, the result is completely irreversible.
This is akin to some of the 'brick paradoxes' mentioned in this blog post, which I found linked in a question while browsing the tag selection a few days ago. There are plenty more to be found in Technic construction; after all, Bryson's Second Law of Repairs, a corollary of Murphy's Law, states: "If you fool around with a thing for very long you will screw it up."
A 1x1 tile stuck to a 1x1 round plate was the hardest, I think harder than the accepted answer above, until I used two LEGO brick spectators in opposite directions. Put the stuck pieces on top of one separator facing north, pry with a second one facing south. Pops them apart beautifully.
Small plates stuck together can be separate using two LEGO separators, but both in the same direction.
If you have 2 brick separators available (one above and one below), or a plate and a brick separator it is easy as you can lock each piece, but with bare hands it is completely impossible.