It's simply not possible to give a hard science answer.
The oldest living people today are a little over 100 years old. So we have no hard scientific facts about life past 100 or so.
If you are going to speculate that future technology may allow a human body to live for millions or 10^80 power years, presumably the same technology could be applied to the brain to also keep the brain alive and functioning. As no one today has any real idea how to extend human life beyond 120 or so years -- there are some theories but they're basically speculation with little or no experimental evidence -- it's just impossible to say how it would work.
Presumably if you speculate that human life can be extended indefinitely, then degenerative diseases would have to be cured. You could cure Alzheimers, etc., so that wouldn't be an issue.
One might plausibly ask if there is some limit to the memory capacity of the human brain. Of course people forget things, but there's no real evidence that we forget because we had to delete old memories to make room for new ones. Maybe that's how it works, and maybe not. No one really knows. Frankly I'd say probably not, as children forget things as readily as adults. If the observed reality was that people remembered everything that happened to them until they turned some age, say 50, and at that point they started forgetting one year's worth of memories for every additional year they lived, then a plausible explanation would be that they had filled up their memory and old memories had to be deleted to make room for new ones. But that's just not how it works in real life.
As the human brain is obviously finite, presumably memory capacity is finite, but just saying "finite" is a long way from saying how much. Could a health person hold 200 years worth of memories? 1000? 10,000? We just don't know.
Would people go insane from such a long life? I don't see why. There's no evidence that living to 100 makes someone go insane. Why would living to 1000 or 1 million? If one was locked in a small room for thousands of years, maybe so. But assuming that one could lead an active life, I don't know of any reason to think long life itself would drive people insane. Of course we have no experimental evidence of someone living to be a million years old. Maybe trying to extrapolate from 100 to 1 million I'm just missing a small trend that would grow very big over time. It's pretty much impossible to say.