A slightly different reasoning leads to the same result without involving infinities:

Let's assume that, say,  x^x^x^x^(x^3)=3, and let x=3^(1/3)

Now for the last part we just have a 3 instead of (x^3), giving a shorter new tower x^x^x^(x^3)=3, and we can keep repeating this process: x^x^(x^3)=3, x^(x^3)=3, (x^3)=3, confirming that x=3^(1/3)

In general, (n^(1/n))^.... (n^(1/n))^(n^(1/n))^n =n