Brzezniak and Zastawniak's book on stochastic processes shows that that there is no way to beat the casino by having a finite amount of money available:
Let $(X_1,X_2,\cdots)$ be independent random variables and let $(\alpha_1,\alpha_2, \cdots)$ be a non-negative bounded sequence of previsible random variables.
$$E(\sum_{i=1}^{n+1}\alpha_iX_i|X_1,\cdots,X_n)=\sum_{i=1}^n\alpha_iX_i+\alpha_{n+1}E(X_{n+1}|X_1,\cdots,X_n)=\sum_{i=1}^n\alpha_iX_i+\alpha_{n+1}E(X_{n+1})\leq\sum_{i=1}^n\alpha_iX_i$$
(Assuming $E(X_{n+1})\leq0$)
Proving that $\{\sum_{i=1}^n\alpha_iX_i\}_{n\in\mathbb{N}}$ is a supermartingale. Therefore there is no possible strategy that can make the game favorable to a gambler.
I also know that if $(\alpha_1,\alpha_2, \cdots)$ was not bounded then the theorem is no longer true. For example, the martingale system where we double the bet each time we lose until we win once has positive mean.
My question is, what is wrong with the previous proof if $(\alpha_1,\alpha_2, \cdots)$ is not bounded?
The process $\{ \sum_{i=1}^n \alpha_i X_i\}_{n \in \mathbb{N}}$ can still be a super-martingale if the $(\alpha_i)$ are unbounded, and will be under the doubling strategy you mentioned, but it may no longer be uniformly integrable. This means that if you only play for a finite time (such as expecting to not live for more than $200$ years) the strategy will be expected to lose money over that time frame, but if you could continue the strategy forever you will make a profit.
The proof can also break down if $\mathbb{E}[|\alpha_i X_i| ] = \infty$ for some $i \in \mathbb{N}$ (say if your strategy involves placing an infinitely large bet at some time). Then $\mathbb{E}[\alpha_i X_i | \mathcal F_{i-1}]$ may not be well-defined, so the proof won't work.