Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?

[i]Proposed by Michael Ren[/i] 
1. **Initial List and Conditions**:
   - Henry starts with the list of the first 1000 positive integers: \( \{1, 2, 3, \ldots, 1000\} \).
   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.
   - After erasing, he decreases every remaining integer in the list by 1.

2. **Understanding the Erasure Condition**:
   - Nonpositive integers are erased immediately.
   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.

3. **Effect of Decreasing by 1**:
   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.

4. **Key Observation**:
   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.
   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.

5. **Erasure of Numbers with Repeated Digits**:
   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.
   - For example, 11 will be erased in the first step, 22 in the second step, and so on.

6. **General Case**:
   - For any number \( n \) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.
   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.

7. **Conclusion**:
   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.

\[
\boxed{11}
\]