At turn 185, I found that when I pressed the "Turn" button, the cooling fans sped up, memory held steady at 8.25 GB of RAM, and the eight CPUs averaged 50% utilization. The AI's turn wouldn't end. This is with an Immense map on Win 7 64. The CPU usage fluctuated, so it seemed the system was trying to do different things and not just the same thing over and over. On several occasions, I let it run in this state for minutes while I did something else. I tried making various changes to the tactical situation, and finally found the resolution.
In a tactical move to prevent a planet from being colonized before my Colony Ship arrived, I had surrounded a planet with six ships, preventing direct access by anyone else. I also have 100 turns where no one can declare war on me. Another race has a Colony Ship two hexes from the planet. This seems to cause the AI to contemplate its navel. If instead I move one ship away from the planet, then hit the Turn button, the Colony Ship makes like a sperm fertilizing an ovum. It's my turn again in just a fraction of a second. From this I conclude that the system is working hard on a solution to the blockaded planet. The developers may want to look at this particular gambit. I think the AI is going down a rabbit hole.