Hydrogen has a critical temperature of 33K. So you can't get a true liquid simply by pressurizing room temperature hydrogen. It would be a supercritical fluid. You can squeeze more into a container with higher pressure, but it won't be a liquid.
Liquification can make transporting some materials easier (like propane, LNG). Once liquified, the materials can remain in this state with sufficient pressure. They can be poured, drain through valves, etc. There is some vapor present, but you can often move the liquid and ignore the amount in vapor.
With materials where the critital temperature is low, you can't do this step at room temperature. Instead of liquifying, the material just gets denser and denser, but still acts in many ways like a gas. There's no fluid portion that you can pour. The fluid will expand to fill a container like a gas.
This doesn't put a hard limit on the density you can reach. You can theoretically make a container of supercritical hydrogen as dense as cryogenic liquid hydrogen. The problem is that the pressures needed to do so are huge. You would need massively strong tanks and valves. Any leak or failure would be very dangerous. You might be able to compress a few grams of hydrogen to the density you want. But at room temperature, the device to hold it at that pressure might have to be several hundred kg or more to not fail. So the engineering side usually means that this is never done. You lose more in the cost and complexity of the necessary equipment than you gain from the extra quantity that can be squeezed into the tank.