As far as I'm aware, the diet with the best track record, backed by data, for helping people keep weight off after the initial weight-loss period is the Rice Diet.
Sixty-three percent of patients kept their weight off for a year after leaving the treatment facility, and I think about 45% regained no more than six pounds after five years (going from memory -- I'll see if I can find the actual stat). For most diets, the commonly cited statistic is that only about 5% of people are able to keep weight off in the long term, though there is
some dispute about how accurate that is. In any case, 45% is really impressive.
(The rice diet is the one where people were originally allowed to eat only rice, fruit, fruit juice, and sugar. Later versions also allowed occasional fish and chicken and maybe a few other things. People were expected to eat this way only while at the facility. Once they went home, they were supposed to eat normally but sensibly.)
There are no long-term stats that I'm aware of for the potato diet or for water-only fasting, but I'd expect them to also do relatively well for the same reason as the rice diet -- namely, blandness. (As with the rice diet, people are not expected to stay on just potatoes or just water for the long term. The point is to use them to normalize appetite so that eating sensibly becomes easier going forward.)
The trick to making it easier to keep weight off in the long term is to adjust your bodyweight set-point, which would otherwise defend against fat loss by ramping up appetite. I'm only a few chapters into the
Guyenet book and he hasn't touched on this yet, but I expect that he'll spend a fair number of words on it given that it's an area of his own research.
I personally recommend chemotherapy for weight loss (though I generally recommend against it in cancer-free subjects for other reasons). I've been losing weight for months despite consciously trying not to.