This infinite exponential tower is known in literate as “tetration” or sometimes “iterated exponentials”, “power tower,” “hyperpower,” or “superexponentiaion.” The details of convergence are fairly complicated, so I encourage you read the award-winning paper Exponentials Reiterated by Arthur Knoebel. Here is a graph that illustrates the domain and range of the infinite power tower.
The infinite tower only converges for 1/e^e ≤ x ≤ e^(1/e), with a corresponding range 1/e ≤ y ≤ e. For positive x values below the minimum value the limit is not defined, and for positive x values above the maximum value the limit goes to infinity.
