To Jinjing-I found your solution

What is (2^0.5)^(2^0.5)^(2^0.5)...?

Your answer is

x^x^x^x^....if convergent, We have x=x^x , Lnx=xLnx, we get x=1,If x>1,It would be Divergent.

But I think it is wrong.  From x^x^x... convergent, you cannot get x=x^x.  Let's say x=.5.  .5, .5^.5, .5^.5^.5,... is convergent because it is increase and smaller than 1^.5^.5..., which is 1.  But you cannot say .5=.5^.5.

Since your reason is wrong, I am not sure that x^x^x^x^....is Divergent when x>1.