We are given a fascinating recursive expression:

$x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 2$

This infinite power tower, also known as a tetration, asks:

What value of $x$ causes this tower to converge exactly to 2?

Let:

$y = x^{x^{x^{\cdot^{\cdot^\cdot}}}}$

Given that $y = 2$ , and because the tower is infinitely recursive:

$y = x^y$

Substitute $y = 2$ into $y = x^y$ :

$2 = x^2$

Solving for $x$ :

$x = \sqrt{2}$

Infinite power towers don’t always converge. For convergence, it’s known that:

The tower $x^{x^{x^{\cdots}}}$ converges if and only if $0 < x \leq e^{1/e} \approx 1.444\ldots$

Check if $\sqrt{2} \approx 1.4142$ is within that range — it is.

Thus, the tower converges and the result is valid.

Final Answer

$\boxed{x = \sqrt{2}}$

Reference