Because if you write down the Bayes' Rule calculation, that's not the ratio that appears in it.
HTTTHHHTHTHTTHHTTHTHTTHTHHHTHHTTTHTH. Using Bayes' Rule, what are the odds I actually got that sequence, as opposed to randomly typing letters? (If you miss my point: You're misusing Bayes' Rule in this argument.)
Nope. They both mean: for large n, for a fraction of sequences that tends to 1 as n -> infinity, that's what happens.
If Alice cheats 100% of the time, your formula produces probabilities greater than 1 for any n less than infinity, which I'm reasonably certain doesn't happen.
Using Bayes' Rule, what are the odds I actually got that sequence, as opposed to randomly typing letters?
Pretending for the sake of argument that I don't see any regularities in your sequence that I wouldn't expect from genuinely random coin flips (it actually looks to me more human-generated, but with only n=36 I'm not very confident of that): the odds are pretty much the same as the prior odds that you'd actually flip a coin 36 times rather than just writing down random-looking Hs and Ts.
You're misusing Bayes' Rule in this argument.
I think you may...
Alice: "I just flipped a coin [large number] times. Here's the sequence I got:
(Alice presents her sequence.)
Bob: No, you didn't. The probability of having gotten that particular sequence is 1/2^[large number]. Which is basically impossible. I don't believe you.
Alice: But I had to get some sequence or other. You'd make the same claim regardless of what sequence I showed you.
Bob: True. But am I really supposed to believe you that a 1/2^[large number] event happened, just because you tell me it did, or because you showed me a video of it happening, or even if I watched it happen with my own eyes? My observations are always fallible, and if you make an event improbable enough, why shouldn't I be skeptical even if I think I observed it?
Alice: Someone usually wins the lottery. Should the person who finds out that their ticket had the winning numbers believe the opposite, because winning is so improbable?
Bob: What's the difference between finding out you've won the lottery and finding out that your neighbor is a 500 year old vampire, or that your house is haunted by real ghosts? All of these events are extremely improbable given what we know of the world.
Alice: There's improbable, and then there's impossible. 500 year old vampires and ghosts don't exist.
Bob: As far as you know. And I bet more people claim to have seen ghosts than have won more than 100 million dollars in the lottery.
Alice: I still think there's something wrong with your reasoning here.