It’s tough to separate the games from their logos. The Munich games are first ones I actively remember. I was five years old back then and even before the tragical events ensued, I thought that the logo, which was hard to escape here in Germany, was great for target practice. It looked somewhat cold, hard and unfriendly to me and didn’t reflect the friendly and sunny atmosphere of that summer at all.
From a purely visual standpoint the Innsbruck logos look the best, most rounded and thought through to me.
Most of the other Designers seem to struggle to include the rings and have them not look tacked on. Of course, the rings themselves are as perfect a logo as can be designed.
Interestingly enough, the Lake Placid games logo does not feature the rings at all.
Since a logo is all about easy and quick recognition even from afar and should work just as well in b/w as in color, many of the oldest ones are great illustrations but bad logos in that regard — and some of the newer ones, too, notably the Atlanta one.
I also like the Moscow logo; it’s clean and elegant and the weight between the rings and the host’s national/city symbol is well balanced, other than with the Tokyo logo, for instance. The Japanese sun weighs down too heavily onto the poor, delicate Olympic rings. At the Sapporo games they did better.