I also found another example that taking a limit

could produce an extra false solution. Your example (2+(2+...)^.5)^.5 is the one.  We will get x=(2+x)^.5.  We get x^2=2+x.  We need to discard -1 as the solution. The reason is the same.  So there is no mistry about original problem.

It is interesting to know that (2^.5)^(2^.5)^...= (2+(2+...)^.5)^.5.