Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?
1. First, we need to determine the position of the letter in the sequence. We are given that Big Welk writes the letters of the alphabet in order and starts again at $A$ each time he gets to $Z$. We need to find the $4^3$-rd letter he writes down.
2. Calculate $4^3$:
   \[
   4^3 = 4 \times 4 \times 4 = 64
   \]
3. Since the alphabet has 26 letters, we need to find the position of the 64th letter in a repeating sequence of 26 letters. This can be done by finding the remainder when 64 is divided by 26:
   \[
   64 \div 26 = 2 \quad \text{remainder} \quad 12
   \]
   Therefore, the remainder is 12.
4. The remainder tells us that the 64th letter corresponds to the 12th letter in the alphabet.
5. List the first 12 letters of the alphabet:
   \[
   A, B, C, D, E, F, G, H, I, J, K, L
   \]
6. The 12th letter is $L$.

The final answer is $\boxed{L}$