&gt; And would such a theory have any practical use?<p>Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function).<p>[1]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Ackermann_function" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Ackermann_function</a>