Dantrag: Neither liquid nitrogen nor dry ice would be a good solution as they are both very temporary, very expensive. and very dangerous to work with.
If you are looking for stable, safe, re-usable "Extreme" cooling, there are four methods that I have seen in use most often.
Method 1 is to water cool (any water cooling kit should work) the cpu and have the water pipes externally cooled, with the radiator actively cooled.
Method 2 is to water cool the cpu with copper pipes as opposed to tubes, and run the pipes through an externally cooled source, such as a water pool that is cooled via an AC, a freezer, or a peltier.
Method 3 is to use a very high powered peltier that has the radiating end outside the PC case that is actively cooled, and possibly have the radiating end water cooled or even in a thermal unit like an enclosed AC. The problem with this method is that peltiers can produce a lot of condensation, especially in high humidity areas.
Method 4, a more clunky but effective way, is to enclose the CPU case inside of a moisture-free thermal unit that is supercooled. You can purchase filters that will remove all the moisture from the air from a unit like a small freezer, and have a shim or something over the cpu/cpu fan. This would do it, although condensation is also an issue here.