The idea of doubling a bet until you win has two major flaws. The one most people focus on isn't even the worst one. That being, you essentially need infinite money (and the casino allowing infinite bets) to guarantee never losing everything.
The downside I find more egregious, is that you only "gain" whatever your starting value was when you eventually win. If your starting bet was 10$ and you lose 5 times, your next bet would be 320$. You would however only be earn a grand total of 10$ if you won the bet.
Example: You lost once, so you're betting 20$. You win, so you get double the amount back (40$). You bet a total of 30$ (10$ from round one plus 20$ from round two). Your total profit was 10$ (40$ minus 30$).
Essentially, you run the risk of losing everything to a slightly unlucky run, just for meager gains.