The upshot is that as long as you only stake $1 at a time, you're unlikely to lose more than $50.
On the other hand, /if/ you do, you'll have to play for 6000 more flips until you can be fairly certain that you're even again.
What's worse is if, after having lost $50, you're down to your last $50, there's almost a 1/5 chance you'll blow all of it trying to recover if you wager $1 each time.
If you grow wise and start Kelly betting you'll get back to your starting $100 on average in 5000 flips, though. If you can take out a loan of $500 first, you can Kelly bet your way to even much faster, in an expected 700 flips. Whether this is worth the interest on the loan depends on how quickly you can find challengers to bet with.
Make a deal with all the banks or systems that can print money that would allow you to take an infinite loan from them. Then just double the bet every time you lose.
If they can print money, why not infinitely as you will always pay it back anyway, so you don't have to worry about introducing inflation. There will always be a point when you can just burn the money that you temporarily introduced.
...from where does the money come that you pay back? Even if you have an unlimited stake, your winnings will be constrained by the counterparty eventually.
Right, I forgot, you also need someone willing to take those bets. So I think what you should do is make bets against multiple casinos/institutions where you can develop an algorithm that will find you an optimal method of betting for reasonable 50/50 results, if it makes easier to think. So for example at some point you might want to go to a casino and put the max bet on a single number in roulette, but do it enough times that you would have 50% odds of winning.
Once that is exhausted, you would have to become more creative, like trying powerball enough times, but I'm not sure how good the odds are there vs the reward. Maybe that wouldn't ever work.
Actually, I forgot. You can just play with highly leveraged options. It's not infinite yet, but come back to me until you've multiplied enough times that even options are not enough.
Forget everything I said before, just play with options and automated algorithm to buy more. And post here once you can't buy any higher cost options, and we'll figure something out together.
On the other hand, /if/ you do, you'll have to play for 6000 more flips until you can be fairly certain that you're even again.
What's worse is if, after having lost $50, you're down to your last $50, there's almost a 1/5 chance you'll blow all of it trying to recover if you wager $1 each time.
If you grow wise and start Kelly betting you'll get back to your starting $100 on average in 5000 flips, though. If you can take out a loan of $500 first, you can Kelly bet your way to even much faster, in an expected 700 flips. Whether this is worth the interest on the loan depends on how quickly you can find challengers to bet with.