Ackermann built a function that is computable but grows faster than any loop can — the first proof that ‘computable’ is bigger than ‘primitive-recursive’ (the functions you can write with plain for-loops). Slide n and watch A(m,n) go from polite to astronomical the instant m ticks up.
A(0,n)=n+1; A(m,0)=A(m−1,1); A(m,n)=A(m−1,A(m,n−1)). The double recursion is why no single loop can bound it. The instrument computes A live for m=1,2,3 and a fail-loud self-check throws unless A(2,2)=7, A(3,3)=61, and the closed form A(3,n)=2ⁿ⁺³−3 holds for n=0..4 — the row that visibly explodes. (m≥4 is left off: A(4,2) already has 19,729 digits.)
We cap at m=3 for a reason — A(4,n) overflows any screen and any Number. That the growth outruns every primitive-recursive function is Ackermann’s 1928 theorem, cited; here we only exhibit the first rows.