My immediate reaction to "Think of a number...Magicians...": The tricks probably require that their targets think of only natural numbers - {1, 2, 3, 4, ...}. And assume that most folks will pick small, dull natural numbers. (Vs., say, 2^24-1. Which is the maximum unsigned mediumint value in MySQL. Or, if you've ever been stuck doing low-level stuff with a 80286 CPU...)
It'd be interesting to see if any of the tricks worked for, say, -7π + ei.
More precise phrasings of the questions are needed for them to work. Even just allowing the rationals makes most (all that I've seen) of these sorts of tricks fail. It could be an interesting problem to find such tricks in the Surreal numbers, since that's the most general form of number (ordered field).
Some obvious ones are the "I can guess your number" tricks, where you do hard math on a secret number and tell the magician the result, who then does easy math to tell you your number.
Such a loop of operations containing only multiplication, addition, and subtraction would work on activity complex numbers. Square roots would not.
It'd be interesting to see if any of the tricks worked for, say, -7π + ei.