What is the best way to dislodge a ceramics espresso cup from a larger ceramics coffee cup?

Try this: phase 1: Make a pitcher of ice-cold water. Make a pitcher of very hot water. Now, pour the ice water into the inner bowl, and let sit for 10 seconds. Empty, and turn over. Now slowly pour hot water on the outer bowl for a few seconds. The trick is to make the outer bowl hot, but not let the inner bowl warm up. Before things cool down / heat up, try pulling them apart. Phase 2: Microwave the bowls for 30 seconds. try to pry them apart. Phase 3: Put them in your oven the next time you put the oven in self-cleaning mode.

Thermal effects are going to be hard because ceramics are very poor conductors. For it to work, you'll need to let it "soak" for a while. Hot water outside the white cup is probably the best option. While the white cup soaks up the heat, you might consider spraying an upside down can of "canned air"  into the blue cup for rapid cooling. You need to keep the blue cup from heating up along with the white cup. My preference is to "airbag" it. Firmly rubber-band a balloon onto a straw. Carefully insert the balloon below the blue cup (using additional straws to push). Blow vigorously into the straw to attempt to air-bag the cup out from below. If this does not work (it probably won't) get more pressure -- an air compressor would be ideal, but a water tap could work as well. Pressure is incredibly powerful in this sort of application, and tap water can produce much more force than your hands can. Caution: this might produce an enormous mess and/or launch the espresso cup into the air. Experiment carefully. Another thought is to pull at angles instead of straight out. If the cup is wedged rather than merely squeezed in there, pulling sideways may be more effective than pulling straight out. My last suggestion is to use a combination of vibration and pulling at the same time. In the oilfield, when pipe threads are stuck, we apply steady torque and then hit the outside of the threads with a hammer. The shock from the hammer breaks the static friction, allowing the straight pull to make some progress. In this case, I would recommend one person pull (hard) on the blue cup handle while the other person firmly taps the back and sides of the white cup with a heavy but non-marring object (like a hardcover book or rubber mallet).

If you haven't had any luck yet with some of the excellent and intriguing suggestions, I suggest turning the white cup over on top of a folded cloth and then tapping the cup with a wooden spoon. When you've made several thousand mugs, you've gotten things stuck in them. And the spoon tap was usually successful. The vibrations eventually shake things loose. Let us know if you eventually manage to keep both cups intact.

1. Try to pass the floss where blue cup is stuck in the white cup. 2. Try #1 while the cups sit on something vibrating - such as bass speakers ( as in Mathijs Booden's idea) Edit: 3. A variation of Ryan Carlyle's "air-bagging" idea. a) Stuff chickpeas (or black gram) under the blue cup, so that they are completely packed. The seeds should be big enough that they just pass through the gap. b) Add some water. Wait overnight. c) The chickpeas would expand and push up the blue cup.

WD40?

Put the white cup in a bowl of hot water with nothing inside then after a few minutes pull. Good luck.

If the espresso cup is especially precious to you perhaps you could break the white cup, this liberating the espresso cup?

Put the white cup upside down and apply some vibration, i.e. from a bass speaker or ultrasonic bath. Wait.

I would tie a string to the handle of the blue cup and try pulling the blue cup around in different angles. To do this more systematically, you can tie a mass to the free end of the string, hold the white cup in the air upside down and let the mass at the end of the string fly in free air, pulling in all directions. Meanwhile you can tap the white cup in various locations. This should free the blue cup after a while. If this doesn't work, you should decide which cup you love the most and break the other. It is the solution with the highest value.