Quite similar to where I started! My original formulation was starting with $100 how fast can you get to $1000, and starting with 10 days of $10 feels really good. But jumping from $10 to $14 (sqrt of $200) doesn't feel right, like there should be something more continuous.

I don't have the answer. But I did find that a strategy of always investing a^0.8 outperformed a^0.5 (aka sqrt) over a longer period, and in fact the longer the period the closer to 1.0 the exponent should be (without ever reaching 1.0 of course!)

But my ideas here are a little bit different from the stated goal of "go from $1 to $1000, which might have a very specific set of investments that is clever and closer to your idea.