The moon rarely gets so thin that it’s invisible. But of course that depends on your eyesight. I can see it with my naked eye even when it’s really thin. Believe it or not, when I used to do it some years ago, I could actually see Venus in the daytime sky. And I confirmed it by then putting the binocs on it. I had to first have some general idea of where it was. If I didn’t have some general idea, I’d probably never find it.
Some years ago there was a contest (reported in Sky & Telescope) to see who could see the new moon the shortest time after astronomical new moon. It was something like 14 hours and I suppose some minutes. That was under the best possible conditions.
I have seen all five of the naked eye planets and have managed to see Uranus and Neptune through a telescope. At that time there were 9 planets. Now that Pluto has been demoted I can say I have seen all the planets in the solar system (but I still want to think of Pluto as a planet).