[jessetemp]

It's not perfect, but pretty close if you compound monthly. It works with your examples too.

100% interest (compounded monthly) doubles at about 0.75 years, and 0.75 interest doubles at around 100 years.

41% interest doubles at around 1.75 years, and 1.75 interest doubles at around 41 years

[feoren]

Compounded monthly? That's rather cherry-picked. 0.75% interest compounded monthly doubles in 93 years, not 100. You're compounding the left side differently than the right side.

When talking about stock market returns, people are talking about year-over-year returns. There is no "compounded monthly"; 41% interest "compounded monthly" is actually 49.7% interest. Everyone would call that 49.7%, except credit card commercials that are trying to trick you.

The point is that GP presented the %-switching thing like a rule, but it's not. They've confused how that rule is applied.

Edit: I tried compounding every second. The rule works exactly. e is just such a magic number that continuous compounding is exactly the point at which this rule becomes true. So the more often you compound, the better this rule is. Wild. I still don't recommend using it for annual percentages.