When you solve an equation like
x^2 = x for x, you usually implicitly specify a field in which the solution lies. Usually the implicit field is the real numbers, but this choice is arbitrary and context dependent. You might choose all the rational numbers AND the square root of two (and all of it's multiples, and it's multiplicative inverse), for example.
Infinity is usually not in this implicit field. As such, your candidate solution, 2^2^2^2 ..., which does not to converge in the reals, or the complex numbers, is not a valid solution.
If infinity is in the field (you could, for example, be in the projective plane), you have to set up rules for how to treat it. As you have demonstrated, these rules would have to maintain that infinity is a solution to x^2 - x = 0.