Is there a pressure at which any gas becomes a liquid, or are some gases genuinely impossible to squeeze into liquid at room temperature?

The reason behind the limit is worth having, because it makes it stop feeling arbitrary.

A liquid holds together because its molecules attract each other enough to stay in contact while still moving. Heat is molecular motion. Raise the temperature and at some point the molecules are moving fast enough that the attraction between them cannot keep them condensed, no matter how tightly you pack them.

Compressing harder does not help because squeezing does not slow anything down — it just puts fast-moving molecules closer together. You end up with a dense, hot fluid rather than a liquid.

Which is why the critical temperature tracks how strongly molecules attract one another. Water, which hydrogen-bonds strongly, has a critical temperature above 370 Celsius. Helium, which barely attracts anything, is around 5 kelvin. Hydrogen is a small, light, weakly attracting molecule, so it sits near the bottom.

Your model is wrong, and pleasingly so — there is a hard limit and it has a name.

Every substance has a critical temperature. Above it, no amount of pressure will produce a liquid. Not a very large pressure, not an absurd one: none, ever. The liquid phase simply does not exist above that temperature.

So the answer for any particular gas is: look up its critical temperature and compare it to room temperature.

• Above room temperature — carbon dioxide, propane, ammonia — and yes, pressure alone will liquefy it. This is exactly why a propane cylinder contains liquid at ordinary temperatures.
• Below room temperature — nitrogen, oxygen, hydrogen, helium — and no pressure whatsoever will do it. These have to be cooled first, and for some of them cooled a very long way.

Hydrogen's critical temperature is around 33 kelvin, which is roughly minus 240 Celsius. Room temperature is not close to that, and the gap cannot be closed with a pump.

Worth adding what you get instead of a liquid above the critical temperature, because it is genuinely interesting rather than a consolation prize.

Push past both the critical temperature and the critical pressure and the substance becomes a supercritical fluid. It is not a gas and not a liquid — it fills its container like a gas, dissolves things like a liquid, and there is no boundary or meniscus anywhere because there are no longer two distinct phases to separate.

The practical importance is real. Supercritical carbon dioxide is used industrially as a solvent, most famously for removing caffeine from coffee beans, precisely because it penetrates like a gas and dissolves like a liquid.

So the phase diagram does not end at the critical point. It stops having a line there, which is a different thing.

Practical corollary for anyone dealing with cylinders: this is why some gas cylinders have a liquid level and others do not.

Propane and carbon dioxide are stored as liquids under modest pressure, so the cylinder's weight tells you how full it is and the pressure barely changes until it is nearly empty. Nitrogen, oxygen and hydrogen are stored as compressed gas, so the pressure gauge tracks the contents directly and there is no liquid in there at all.

Same steel bottle, two completely different relationships between gauge and remaining contents, and the critical temperature is the whole reason.