This is beginning to come close to the sort of thing I was trying for in my failed attempt. It was clear that for probabilities near zero "twice as likely" meant roughly doubling the probability, and clear that for probabilities near one, "twice as unlikely" meant roughly getting twice as far away from one; and then it occurred to me that a layperson might naively expect "twice as likely" and "twice as unlikely" to be inverses under composition (so that "twice as likely" as 0.99 would be around 0.995). So I searched for a function which had gradient 2 near zero and gradient 1/2 near 1, and then I generalised to "k times as likely" and searched for a family of such functions, so that f_k had gradient k at zero and gradient 1/k at 1, and such that f_k composed with f_m was f_(km).

Hmmm. The "twice as likely" one seems wrong in the middle somehow, if twice p=0.5 is "just a bit more than that", the function would sort of accord, but restricting the usage seems as good an answer.

Of course, I need to go and find some suitable functions Fs anyway for my own satisfaction.

That was my eventual conclusion, yeah. I found functions which had all the properties I'd aimed for, but they behaved so unhelpfully in the mid-range that I despaired of getting anyone to even begin to agree that they were a good definition of the phrase "n times as likely" in the general case :-)

Which is always very unsatisfying to write up, isn't it?

I think I will leave this for more productive persuits ;)