Does ((2^2)^2)^2........ = 1 or 0? let $x = (((((2^2)^2)^2)^2)^2)^\cdots$
Therefore, $x^2 = ((((((2^2)^2)^2)^2)^2)^2)^\cdots$
But this is the same expression.
Therefore $x = x^2$
Therefore, $x^2 - x = 0$
Therefore $x(x-1) = 0$
Therefore $x = 0$, or $x = 1.$  
Where is the error in this argument?  I assume there is an error as repeated power of $2$'s should make the number larger and larger.

$\infty$ is also a solution of $x^2 = x$, since $\infty^2 = \infty$. It is the correct solution in this case.
