Given a biased coin with probability to come up heads as $p$ and a fraction of your money to bet each time as $q$, what is the optimal $q$ to grow your money by $m$-fold in the least amount of bets?
Given that $m$ is sufficiently large (to ignore weird finite size effects), I've sampled this distribution for various values of $p$ and $q$. Here I've used $m=50$ and $10,000$ trials.
Is there an explicit formula for the minima of these curves?
The number $X$ of bets required to increase your money by a factor $m$ is of course a random variable, so it doesn’t make sense to ask how to minimize it. I gather from the axis labeling in your graph that what you actually want to minimize is the expected value of this random variable.
It’s going to be hard to do this exactly, since $E[X]$ is not a continuous function of $q$; it jumps whenever the result of some number of wins and losses crosses the threshold $m$; so we can’t minimize it by differentiation. However, if you don’t want to perform the complete discrete optimization, a good approximation is to minimize the time at which the expected value of your wealth reaches $m$. The logarithm of this expected value increases by
$$ p\log(1+q)+(1-p)\log(1-q) $$
in every bet. Setting the derivative with respect to $q$ to $0$ yields
$$ \frac p{1+q}-\frac{1-p}{1-q}=0 $$
and thus $q=2p-1$, which roughly agrees with the minima in your graph.