Chaingun's "set as message in log instead of pop up window"-list:
1. Most or everything regarding warfare should be moved to logs because the sheer quanitity of these messages can kill one (people already discussed these in detail above I think).
2. Stability and technology increase/decreases have no use as pop ups (always pay attention to log though so you don't leave treasury at max when getting to +3 stab).
3. Merchants. I guess this depends a bit on what you prioritize. I find it helpful to know whenever I'm competed out, but then again this is not a priority during war. Also, all other messages regarding successful and unsuccessful trade expansion attempts should be logged only.
4. These promotion and demotion boxes are totally uninteresting (I'm talking about those that trigger based on certain VPs). Remove them totally; there is no need to clutter the log.
5. Building contruction finished notices are good log candidates.
These are all I can think of right now. Most of them are no-brainers except the warfare ones. The "fleet has faced too high attrition and is return to port" message should be disabled, because I sometimes get spammed by 20-30 of these at the same time when a fleet I totally forgot about somewhere dies. It's terribly annoying and nothing one needs when he focuses in a war.