It depends on your situation. Some conditions require more material than others. Obviously an 8:1 low power engine with low cylinder pressure can live thinner than a boosted 14:1 piece. I think for general street use, 500-ish hp, that .120 is a good number. I had CP thicken up the pistons we made for you Mr. Jay, because I had endurance in harsh conditions on my mind. It also depends on the area and the "area" of the thin spot. I wouldn't do it myself, but I have taken apart some engines that had been raced hard for lots of runs, and seen .070-.080 thick in small areas.
I also remember Kip Martin pulling a head once for teardown, and the flat tops had turned themselves into dishes! I was young, and I didn't ask him how thick those were, but safe to say.....he had gone too far!

Generally, the piston folks are not scared to go .150-.180 on NA pistons for just about anything. Sometimes, when a dish is really deep, and you are running out of room in the forging, things get tried, and most of the time it works as long as the tune is good.