May be you have seen it. Thanks anyway.

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

Here is a solution.

Let it be x, then we get x=(2^0.5)^x. It seems it will lead x=2 or 4.

The limit should be unique.  What is wrong with this method?

(Someone said that if x^x^x... (x>0) has a finite limit, x should
be <=e^(1/e).  If he was right, then the maximum limit of x^x^x...
is e.  So 4 cannot be the answer and 2 is.  Is there a way to get
the answer directly (which is 2) and not use the maximum limit e to
reject 4?
)