Log-odds aren't what probability is, they're a way to think about probability. They happen not to work so well when the probabilities are 0 and 1; they also fail rather dramatically for probability density functions. That doesn't mean they don't have their uses.
Similarly, Bayes's Theorem breaks down because the proof of it assumes a nonzero probability. This isn't fixed by defining away 0 and 1, because it can still return those as output, and then you end up looking silly. In many cases, not being able to condition on an event with probability 0 is the only thing to do: given that a d6 comes up both odd and even, what is the probability that the result is higher than 3?
[I tried saying some things about conditioning on sets of measure 0 here, but apparently I don't know what I'm talking about so I will retract that portion of the comment for the sake of clarity.]
In more mathematical settings, you can successfully condition on events with probability 0 (for instance, if (X,Y) follow a bivariate normal distribution, you might want to know the probability distribution of Y given X=x).
You can't really do this, since the answer depends on how you take the limit. You can find a limit of conditional probabilities, but saying "the probability distribution of Y given X=x" is ambiguous. This is known as the Borel-Kolmogorov paradox.
I was very interested in the discussions and opinions that grew out of the last time this was played, but find digging through 800+ comments for a new game to start on the same thread annoying. I also don't want this game ruined by a potential sock puppet (whom ever it may be). So here's a non-sockpuppetiered Irrationality Game, if there's still interest. If there isn't, downvote to oblivion!
The original rules:
Enjoy!