My best experience is somewhat similar to Chewey's. Back in SEIII, I started out in a system at the end of a long, long string of systems. My empire grew along its length, without encountering anyone else along the way; I had plenty of opportunity to build up nice infrastructure and defenses, and just about the best defensive arrangement of warp points one could have (an enemy would have to slog his way through my systems one after the other, in sequence). At one point a spatial anomaly threw one of my colony ships to an unoccupied system somewhere else in the galaxy, and I promptly set up shop, but that happened just before I finally reached the end of my system chain.
There was an AI-occupied system right at the first intersection blocking my expansion. First ones I'd met and it was already well into the game. I tried opening peaceful contact, but they refused my offers and attacked like slavering Zebrankies.
Boom... boom... boom... A massive AI warfleet was coming down my corridor of systems, slamming its way methodically through every layer of defence and exterminating every planet along the way. I was stunned; I'd been building these defences and systems for a long time now, and the only thing slowing the enemy down was the fact that it was a long road for them to travel. I got increasingly desperate as my linear empire burned down like a fuse. Finally, they tore into my home system at the end of the line, where all my non-combat ships had retreated to in a massive exodus fleeing their advance. Cornered, trapped, with billions of refugees in the cargo holds... no mercy. The slaughter was incredible. When the Last planet died under the whithering barrage, all contact with my exterminators and their allies was lost.
Leaving only a stunned and horrified population of four million colonists stranded out in some unknown distant corner of the galaxy on their little rocky colony world. I quickly discovered that that system was one of five belonging to a small disconnected group, explaining the loss of contact with everyone when the Last planet in my original empire was lost. Since it was still relatively early in the game, I didn't have stellar manipulation tech yet; I was stuck with what minimal resources I had right there. But conversely, that meant that my enemies couldn't get to me either. They didn't even know any of my people had survived; they probably thought their great genocide was now their own little secret, never to come back and haunt them.
It would. Oh yes, it would. I threw myself into research, working my way to the top of stellar manipulation with a monomaniacal fervor, and built and mothballed a huge supply of warship hulls for use after the eventual breakout.
Which I eventually did. I opened a warp point to the same system I had first encountered that AI, blew it away with a starkiller, and then began putting _it_ through the same meatgrinder it had put my people through. And once I was done with it, I exterminated its allies. And then everyone else in the galaxy, just for good measure.
I love it when my people go insane with vengeful rage. I usually play such a "nice guy" race that I never get to do stuff like that.
My worst experience was probably the time when, having conquered an entire SEIV galaxy except for one race in one heavily fortified system in a remote corner of the stars, I decided to switch over to playing that race against my now-victorious one for a real challenge. One turn after switching, I am informed that the star of my one system will go supernova in 3 years. Not much I could do at that point but evacuate as many people as I could and then gift away the transports in hopes they'd be resettled someplace safe.