I will throw a cat into the bag,
As you said, you don't have a lot of knowledge into what increasing compression ratio does, so don't leap into increasing it!
What has been said so far is correct, but what also is missing is some of the down falls.
Increasing compression can cause pinging or detonation if it is too high or other conditions are not optimum, then the need for high octane fuels to prevent this, otherwise engine failure is likely to happen.
High compression engines can be harder to start.
High compression engines can have less engine life.
High compression can sometimes increase engine temperature.
But as always, there is always pro and cons to anything. :s5