The most common way to increases compression is with new higher compression pistons. But there are other ways to make smaller increases like, milling the head, thinner head gasket, flat faced valves. These are normally used to bump compression up in small amounts. Lets say to have a 9.2:1 C/R and you want to increase to 10.5:1. Swapping pistons would be the best way to do this. As the other methods only increase C/R by a little, but it does all add up.

 

^ Pretty much everything I said LOL. And milling the head/block and thin headgaskets bump it up a lot more than you'd think. But obviously pistons are going to be the biggest jump. Streetable n/a high comps are usually around 12.5:1. High compression but still is a good streetable car. I've seen Honda's in to the 14+:1. I saw an H22 where they even welded in the ehad of the combustion chamber to fill it in more to shrink the combustion chamber. The was like 15:1 compression I think.

When you bump up your compression you will have to compensate for the fuel being compressed more and buy higher octane, which is going to be premium. The higher the octance the better, some gas stations have higher than others.