As a simple counter-example to your claim, the sequence:

  x, x^x, x^x^x, x^x^x^x, ...

when x = sqrt(2) is strictly increasing and bounded above, and therefore converges. It's not hard to show that it's bounded above by 2, because x^x^2 > x^x^x, and x^x^2 = x^2 = 2. Repeat for any length sequence of exponentiation.