1. **Problem statement:** We want to find the largest number of chicken nuggets that cannot be purchased using packs of 6 or 13 nuggets. 2. This is a classic example of the Frobenius coin problem, where we seek the largest integer that cannot be expressed as $6a + 13b$ for non-negative integers $a,b$. 3. Since 6 and 13 are relatively prime (their greatest common divisor is 1), the largest such number is given by $$6 \times 13 - 6 - 13 = 78 - 6 - 13 = 59.$$ 4. Therefore, the largest number of nuggets that cannot be purchased is **59**. **Final answer: 59**