For the first generation, the probability of extinction is 1/4.
In the second generation, the probability is (3/4)(1/4)(1/4), which is the probability that both children fail given that the first generation was successful.
For the third generation, it is (3/4)^3*(1/4)^4 + (3/4)(1/4)(3/4) * (1/4)^2 + (3/4)(3/4)(1/4) * (1/4)^2.
Since it's getting messy, there's probably a better way to organize things, and possibly you can get an exact result somehow. (I'm reminded of those infinite sums that turn out to have in the answer, although I have no idea what it would be in this case.) But if you only need an approximation, you can get a pretty small error quickly this way, just on paper. And with a computer program you can get to a high number of generations quickly using dynamic programming.
Oh and since the bacteria reproduce asexually, we don't need to consider the possibility that they take varying times to decide to reproduce. Whether they wait a minute or two months, the overall result is the same. But I am assuming that the bacteria have limited lifespans, and so they all attempt to reproduce.