In addition to the other good examples that people have
mentioned:
The "Twelve Coins" puzzle is a fun and easy-to-understand
example that thoroughly refutes the idea that you should
never vary more than one variable. http://www.av8n.com/physics/twelve-coins.htm
The "Twenty Questions" game (if you do it right) can also be
seen as very relevant to this issue, and is even easier to
analyze than the Twelve Coins puzzle. http://www.av8n.com/physics/twenty-questions.htm
=================
On the other edge of the same sword, it is possible to come up
with examples where it does make sense to vary only one variable
at a time ... in particular when the data is cheap and the cost
of just plain /thinking/ about the data is dominant. http://www.av8n.com/physics/scientific-methods.htm#sec-doe
So, as usual, it doesn't pay to get too dogmatic in either
direction.