What is the maximum number of variables in a properly designed experiment?
There are an infinite number of variables in every real experiment!
This is kind of a philosophical question, so I'm going to be a little philosophical.
The purpose of an experiment is to measure the effect of one or more independent variables on a response. For instance, we could measure the force we exert on a piece of wood and measure the deflection of the wood as the response. In an ideal world there is only 1 variable in this experiment.
But this is not reality. We are only controlling one variable in this experiment. There are still a lot of variables we are not controlling. What about the age of the wood? Where did the wood come from? What about the ambient humidity and temperature during the measurement? What about how the wood was cut - with or against the grain? Plus there are a lot of variables that you THINK don't matter - the time of day, the temperature 100 miles away, the date of your birth, etc.
Well-designed experiments try to control all of the independent variables that the experimenter THINKS matter. Of course, the experimenter cannot possibly and completely control all variables that DO matter. This is why there is always error in every empirical experiment. Error is caused by uncontrolled variables in an experiment that make a difference to the outcome of the experiment. Even PROPERLY designed experiments have error. In fact one USES the error in the experiment to determine if the variable(s) under investigation have a statistically significant impact.
Another question would be how many controlled variables should there be in an experiment. Statistically speaking, there really is no limit to this number provided that one is willing to do enough experiments. In general, I like to have at least twice the number of experiments as the number of parameters I am trying to estimate. This allows me to estimate my error and differentiate signal from noise.