The martingale is probably the oldest roulette system in the world. Its simplicity means that even a 5 year old could work it out. Basically, you bet 1 unit. If it wins, great. If it loses, you double up your bet. If that wins, great, you win your first loss back and 1 more unit. If the second bet loses though, you double up again. Keep going until you win a bet which will wipe out all your losses and make you one unit.
In theory, this will work. In practice though, it will not work, else we'd all be sat on a beach playing roulette.
Here's a video of roulette extreme system designer running the martingale with unlimited bet size and unlimited bankroll. You can see after 10000 spins, it did make a profit. Early on it hit a run of 13 non-reds in a row and would have hit the table limit in most casinos.
You can run this forever and you will always come up on top, as your unlimited bankroll will just cover your unlimited loss. No one has unlimited bankroll though and no casino will accept an unlimited bet so it's more of a theoretical exercise.
This one looks like a good case, but if you look at the statistics, the max bet needed was £4096 to recover. So 12 losses in a row occured.
Over a run of 10000 spins, these long strings of losing bets will occur. You might think that the chances of losing 10 in a row is tiny, which it is. Its like 0.1275 [(19/37)^10]. Over 10000 spins though, it's quite likely to happen.