Why are samples always taken from iid random variables?

A random variable is something that has one definite value each time you do the experiment (whatever you define "the experiment" to be), but possibly a different value each time you do it. If you collect a sample of several random values, the production of all those random values must -- in order to fit the structure of the theory -- be counted as part of one single experiment. Therefore, if you had only one variable, there couldn't be any different values in your sample.