What's the real life purpose of Knuth arrows? I recently read about Knuth's Arrows. Didn't even know those operations existed. My questions is: Do they have real-life applications? Most of the times a mathematical development follows a real-life need (or another science need). What's the originating need to conceive these hyperexponential operations?
Edit: By REAL LIFE (or another science) NEED I mean a need which derives from a real-life application (e.g. say integration originated from trying to guess circle area, and derivatives solve many economics/physics problems, while simple polynomials have several applications including odds and computer science...), either directly or indirectly wrt solving the need.
 A: The Knuth arrow notation is for expressing primitive recursive functions at a specific level of the hierarchy of such functions. Level is the depth of nested FOR loops needed to compute the function.
The most frequent (but still quite uncommon) use of the notation would be to graphically display how inefficient a particular upper bound argument is in a combinatorial proof.  The numbers connected to Graham's paper with "Graham's number" are an example.  The actual value of the quantity bounded in the paper is probably much smaller, like $13$ or $20$. 
I have never heard of a genuine lower bound like $n \uparrow \uparrow \uparrow \uparrow n$ (something larger than a tower of $n$ exponentials, but still primitive recursive) appearing in a combinatorial problem.  If it did, it would mean that something like Knuth notation is unavoidable when considering that problem.  
There are natural problems in computer science where functions beyond the primitive recursive hierarchy can appear, but strictly speaking these are not covered by the Knuth notation.  If you allow extensions like $a \uparrow \cdots \uparrow b$ where the number of arrows is a variable quantity $n$ then Ackermann-like functions can be encoded.  Conway's generalization of the Knuth arrows includes that.
