Pressure
How much pressure are we talking about here?
Well, it varies a lot. Anything from a few psi (say, 0.1 bar) to 45+ psi (3.0+ bar) in competition engines. Here I'm talking about the increase in engine induction pressure above normal atmospheric pressure. This is what's known in industry as "gauge pressure" or in automotive circles as "boost pressure".
Another way to look at is in absolute pressure and this actually gives you a better picture of how "super" it is. Absolute pressure is the measure of pressure above absolutely no pressure or a complete vacuum, such as in outer space. As mentioned earlier, normal atmospheric pressure at sea level is about 14.7 psi or 1 bar or 1000 millibar or 100kPa, depending on your preferred unit of measurement.
So, let's say we have a boost pressure of 7.5psi (or about 0.5 bar). This is around 22.5psi absolute or 1.5bar absolute and is therefore about 50% above normal atmospheric pressure. So we could reasonably expect our engine to be receiving almost 50% more air than a NA engine. And if the correct amount of fuel were combusted in this air we could expect the engine to produce about 50% more power, yes?
Well, yes, that's true in theory but in the real world there's no such thing as a free lunch and we have to 'pay' for this extra pressure and power somehow, so the actual power gain will always be less than 50%.
Similarly, a boost pressure of about 15psi or 1.0bar (that is, about 30psi or 2.0bar absolute) will give a large power increase over the same sized NA engine but it won't be anything like a 100% increase.
SI Engines
Spark ignition engines for normal road use generally use relatively low boost pressures, often 6psi (0.4bar) or lower. Higher boost in SI engines can be difficult to manage, due to problems controlling pre-ignition, although there would be many competition engines running far higher boost levels.
CI Engines
Compression Ignition engines do not have the same issues with pre-ignition as they ingest only air. They are therefore more naturally suited to forced induction and the amount of boost pressure is really limited only by the design and strength of construction of the engine. Many production engines are designed to run at least 15psi (1.0bar) of boost (for example, the Nissan ZD30, Land Rover Tdi and Td5 engines etc.). I've read of modified American light truck diesels (Dodge Rams, Ford F-series etc.) running in excess of 30psi (2.0bar) boost, apparently with satisfactory reliability(?)