It's not a paradox at all. It's simply not mathematical. The first question, which is completely ignored:

Is x^x^x^... well defined?

And does he mean x^(x^(x^...)) or ((x^x)^x)^x...

Because of his second statement, I guess he means x^(x^(x^...)). So he is looking at the limit of the sequence (x, x^x, x^(x^x), ...). So you have the function f

   x |-> limit of (x, x^x, x^(x^x), ...)

This function is not defined for all x (e.g. it diverges for x>=2) and (thus) its target set is clearly not R.

You can see that it does converge for x=sqrt(2), because 0 < sqrt(2)^a <= a for all a in 0..2) and thus it is correct that f(sqrt(2)) = 2 (by using his statements).

You can also see that 4 is not contained in the target set, because a^4 > 4 for all a>sqrt(2) and because f is monotone for x>0 and because f(sqrt(2))=2.

Thus, the set {x | f(x)=4} is empty.

I.e., his assumption that there exists such an x results in the wrong conclusion.