Maximally touching toruses
7 identical cylinders can mutually touch each other, if sufficiently long. For cylinders of different sizes, 8 can touch each other.
What is the maximal number of mutually touching toruses? I suppose it might be 7, using the 7 cylinder solution. But perhaps there is something more clever.
Just to answer the comment-question by "bubba" with a picture:
(Image from http://www.mathpuzzle.com/)
And see the paper "On the Number of Mutually Touching Cylinders" by K. Bezdek: Combinatorial and Computational Geometry 52 (2005): 121-127. (PDF download)
Update.
The cylinder problem has just been solved for $\infty$-cylinders
(Aug 2103):
"Seven mutually touching infinite cylinders."
Sándor Bozóki, Tsung-Lin Lee, Lajos Rónyai. arXiv:1308.5164: