One of the most irritating, annoying parts of this game is that at the end of nearly every turn, I have a horde of pop-up dialogs to click through. There's always one bit on each, and I've had over 8 'click actions' between turns just to get to the next turn.
Amongst other things, I'm informed of such trivially useless bits of information, like that a mining starbase belonging to the Korx just defected to the Torians (this happens at LEAST once a turn by the midgame). Let me be frank - it's not MY starbase, I have absolutely NO reason to care.
Another annoying screen are the 'war' and 'truce' espionage popups, like "Our agents have reported that X is paying tribute to Y", or "Our agents in the Y government have reported they have declared war on X."
First of all, I don't need a special popup about my AGENTS telling me who declared war and who slept with whomevers mother. I'll get that information on the news screen anyway. Secondly, if I have no espionage, why am I getting these reports?
Another thing are anomaly reports. Turning them off leaves you TOO blind, as you have no idea where your flagship ran off to, if it's still working, or if it's stuck. However, leaving them on bombards me with worthless discoveries (1bc, for example) and most infuriatingly, it centers my screen on the flagship every time (which should definitely be fixed).
It sometimes takes longer to get through the endless wave of popups than it does to perform actions on my turns. With all the endless dialog boxes popping up unnecessarily, I feel like I'm using windows Vista with user control still enabled.
However, about the information overload, I propose this - the greatest idea ever for this game.
Each section can be expanded or collapsed (and if there is no information, it's not even there). Economic Situation contains all those pop-ups about new trade routes, canceled trade routes (that'd be headline news, actually), and mining starbase reports.
Colony news can contain minor information, like if a planet has an overpopulation problem, a report will show up in there letting you know, as it's easy to miss something like that.

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Vaguely unrelated jibba-jabba:Furthermore, I don't want to, nor SHOULD I be told, that Race X is going to war with Race Y because the Drath paid them off. Let there be some mystery to the galaxy.
1) Unless I am friendly with a race, it sort of kills immersion a bit that they tell me every part of their political game.
2) When I haven't even DISCOVERED a race, I get popups from them about who they're going to war with.
3) Espionage should dictate all this as well. If I have no espionage invested in the Drengin, I should never know who is paying them tribute, and how much. I shouldn't know about the wars they are in either. I shouldn't know about their mining bases or their planets flipping.
Honestly, I don't think we should be able to see anything on the 'reports' screen either. We should be able to see who has wars going on the Treaty screen, but not their treaties or trade, not without some espionage invested.