Find a threshold such that one function is always bigger than the other
As you suspected, the threshold is indeed 3.

Here is one approach to prove that for all numbers $y \ge 3$, $c(y,y) &gt; d(y)$.

Step 0, show that $c(\cdot, \cdot)$ is strictly increasing with respect to each variable when the second variable is not 0.
Step 1, show that $c(1,n) \gt n^2$ for all $n\ge0$.
Step 2, show that $c(2,n) \gt 2^n$ for all $n\ge0$.
Step 3, show that $c(3,n)\gt d(n)$ for all $n\ge0$.
Step 4, show that $c(y,y) \gt d(y)\text{ if }y\ge3$. 



Exercise. Show that $c(3,y) &gt; \underbrace{2^{2^{\cdot^{\cdot^{\cdot^2}}}}}_{2y+2 \text{ copies of }2}$ for all $y\ge0$.