You finish your degree, and start your PhD. The first year, you are busy learning techniques and getting caught up with the relevant literature. You are far too concentrated on learning new things to get any thinking done.
In your second year of your PhD, you are getting better. You can do most things without thinking about them. This frees up your brain to think about other things. However, your grasp of the wider literature is still lacking, so you use that brainspace to optimise your current experiments (as you should).
In your third year of your PhD, you are starting to write things up: either your thesis, or your first (big) paper. You read a lot more, you know a lot more. The deep thinking can commence.
Your first postdoc is probably your most productive time: you know what you are doing; you know the state of the literature and which parts are reliable and which aren't; you have a clear idea of what problems need solving. You are starting to write your first grant applications, but you only need one for yourself and not several to cover the needs of a full lab. You don't have any kids at home. This is a good time to solve some big problems. It lasts about 2-3 years.
At the start of your second postdoc, you panic. The big problem was harder than you thought and you don't have enough high-impact papers to be competitive in job applications for a principal investigator (PI) role. You start churning out low-value fillers and collaborating with everyone and their hamster to get your name on as many papers as possible. The rest of the time is taken up by applying for grants and PI positions. You don't even make it to the interview stage. You start pondering about life outside of academia.
The big problems are forgotten.
Curious about the extreme cases. Did any centenarians ever managed to come with an outstanding original math result? If it didn't happen before, I hope to see it happening in the next decades, given current demographic trends.
I was in the relevant department at the time (Harvard math), but I wasn't much of an algebraic geometer, so I took that at face value without probing for details.
Teens? No way, not really ever.