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DanielLC comments on Log-odds (or logits) - Less Wrong
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It seems to me that this doesn't have any real advantage over odds ratios. If I want to do a Bayesian update, I multiply the odds by the relative likelihood. In the example in the article (1/10,000 chance of having the disease, 3% false positive, and 1% false negative), You just take 1:9999 and multiply it by 0.99/0.03 = 33:1 for each successful test. Then you have 33:9999 = 1:303, then 33:303 = 11:101, and finally 363:101 for the final test. Then to change back, you just take 363/(363+101) = 78.23%. The calculations are slower (two multiplications vs. one addition), but it's much easier and more intuitive to convert between them and traditional probabilities.
What you've described is in fact, exactly the same thing as log-odds - they're simply separated by a logarithm/exponentiation. Thus, all the multiplications you describe are the counterpart of the additions I describe. I agree, we could work with odds ratio, without taking the logarithm - but using logarithms has the benefit of linearizing the probability space. The distance between 1 L% and 5 L% is the same as the distance between 10 L% and 14 L%, but you wouldn't know it by looking at 2.72:1 and 150:1 versus 22,000:1 and 1,200,000:1.