[Huskiez]

But no matter what your bankroll is, there's always a chance that you lose x consecutive bets such that you end up with nothing (x increases as you increase your bankroll), and because you're placing this bet infinitely many times eventually that probability (no matter how small it will end up to be) will occur? That is just my reasoning.

In your example the probability of my losing 100 times (thereby busting me) in a row is 1/(36^100), which will happen rarely, but can happen. If I win my first bet, the probability of my busting is 1/(36^101), which will again happen rarely, but can happen (etc).

I think of it like one bet with all of your bankroll, where the chance of success is insanely high (1 - (1/36^100)) in your example, but because there's a chance of failure not equal to zero (here 1/36^100), you're going to end up broke.