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Liquid helium has a boiling point of around -269°C and is used by overclockers. It has led to various world records (e.g. see here and here). What cooling technology can beat liquid helium for the purposes of cooling an overclocked CPU?
(I know dilution refrigerators are routinely used for quantum circuits but I don't know if they can be used for larger circuits like CPUs.)
Short Version
There are several methods of cooling below -269°C, but they won't be much use for overclocking a commercial CPU, because coolant temperature isn't everything.
Cooling Options
Instead of using centigrade, people use Kelvin when talking about very low temperatures. That is becasue there is a minimum possible temperature which can exist, called absolute zero. So it makes sense to say how close you are to it. Kelvin just means "degrees centigrade above absolute zero".
Liquid helium at atmospheric pressure: 4.2K, or -296°C, lots of power.
Liquid helium at almost vacuum: 0.7K, lots of power, but much more expensive
Liquid helium-3 at almost vacuum: 0.3K, probably a watt at most, ludicrously expensive to buy the helium isotope.
Adiabatic demagnetisation of a salt: about 0.1K, a few milliwatts, not too expensive but only works for some hours then needs to be reset.
Dilution refrigerators: about 0.01K, less than a milliwatt, difficult and expensive
Nuclear demagnetization: a few nanokelvin, but only a microwatt or two of power.
So you can probably tell right away, most of these options don't provide anywhere near enough cooling power to be useful for overclocking.
Temperature isn't everything
When overclocking, the aim isn't actually to get the core as cold as possible. In fact most silicon devices give up about -50°C/220K. Instead, the aim is to keep the silicon junctions cool. To do that you have to get the heat to flow out through the chip, and into the surrounding coolant, then carry it away. The heat will flow faster if there is a bigger thermal gradient through the chip, so that is why it helps to get the outside of the chip really cold. But the heat flow is very roughly dependent on the temperature difference. So going from 300K-4K=296K to 300K-0.1K=299.9K is not actually all that beneficial, only about 1%.
Instead, what is important is making sure that there is plenty of cooling power, and that the heat flowing out of the chip is carried away by the coolant. In reality, no matter which of the cooling methods you choose, you probably won't get the surface of the chip down to the temperatures described above. So it's well worth trading off a bit of theoretical best temperature to get a better way of transferring the heat.
Helium is good because although it doesn't take much heat to evaporate it, the resulting gas is very cold, and has lots of heat capacity. That cold gas will flow over the chip as it boils and carry away lots of heat. The other options don't really produce that lovely gas, so won't be an improvement.
Nitrogen ought to be better than it is, but has something called the leidenfrost effect. The evaporating nitrogen gas blows the nitrogen liquid away from the thing you're trying to cool. So you don't get as much cooling as you'd expect. I'm not 100% sure, but I think most of the other low boiling point liquids are the same.
So helium will be hard to beat. Better heat exchanger designs might be worth some effort though.
Liquid Helium is a very poor coolant unless you really want to go to absolute zero. It's heat of vaporization is very low compared to LN2. Reading through the table of the gases, you might find liquid Xenon is better although it is vastly more expensive.
Heat of vaporization is important because it means you need to circulate less fluid before it starts boiling. The difference in flow rate between He and Xe is about 100x
