The goal is to get more power. What would happen If I raised the compression ratio of a 2-stroke. For instance I have a Zenoah 41cc from a back pack blower. It runs fine on Coleman, don't want to start a flame on that, but it shows the compression is very low. Coleman is about 40 octane. So if I flycut the bottom of the jug, say, about 100 thousands and domed the piston to match I would raise the CR some. I would have to raise the ports about 100 thousands too. I have a good lathe and a mill to do these mods accurately. Hypothetically if my goal were 10:1 I would then need to run hig test gas, but that is no big deal. Would I gain any significant power increase? The engine is a pig anyway, but, would I ruin it if I tried?

All good theory, but make sure you check the piston clearance before you go whacking away at the cylinder.

That is the idea of domeing the the piston by the amount I take off the jug. Not really doming it but cutting a relief around the perimiter. By doing so the the squish clearance is maintained

Just cut the cylinder untill the squish is tight as you want it. Thinning the top of a piston may not be good for engine life.
Maybe just measure the base gasket , leave it out and cut the ports to match. Then see what happend.

In reply to original poster:
A lathe is a very handy tool, but you need a 4-jaw to get the cylinder to run true. Then it is easy to adjust the combustion chamber to the piston dome and get the right squish of 0.5-0.8mm. Most chain saw engines have a larger squish to steer clear of mechanical interference.
Remember that lowering the cylinder also means you have to raise the exhaust port. Lowering also means that transfer is reduced, so the engine becomes more sensitive to exhaust system. Using canisters instead of the garbage side dump systems (which BTW can be quite sophisticated) would call for some extra exhaust port raising. The increased blowdown is a free bonus for better power, again IF the muffler takes advantage of that.
Regarding high CR:
It is not always advantageous. Better gas flow is more important for mean effective combustion pressure. Raising the CR just raises the power by some 2 - 5% whilst very much shortening engine reliability. Improving the engine basics without raising CR can easily bring 25% extra power without reduction in engine reliability.
The sad thing is, that rubbish in, is rubbish out. Starting out with a bad engine is not the best idea, and probably will cost quite a lot more than investing in a better engine right away.

I agree 100% with Pe's and Terry's (tkg's) post. Upping the C/R will gain a small amount of power but there are better ways of gettting the increases. Good luck and have fun.