A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number?

[i]Author: Ray Li[/i]

[hide="Clarifications"]
[list=1][*]Lucky numbers are positive.
[*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][/hide]
To find the 17th smallest lucky number, we need to list the lucky numbers in ascending order. Lucky numbers are numbers whose digits are only 4 or 7. We can generate these numbers systematically by considering the number of digits and the possible combinations of 4 and 7.

1. **1-digit lucky numbers:**
   - 4
   - 7
   There are \(2^1 = 2\) lucky numbers with 1 digit.

2. **2-digit lucky numbers:**
   - 44
   - 47
   - 74
   - 77
   There are \(2^2 = 4\) lucky numbers with 2 digits.

3. **3-digit lucky numbers:**
   - 444
   - 447
   - 474
   - 477
   - 744
   - 747
   - 774
   - 777
   There are \(2^3 = 8\) lucky numbers with 3 digits.

4. **4-digit lucky numbers:**
   - 4444
   - 4447
   - 4474
   - 4477
   - 4744
   - 4747
   - 4774
   - 4777
   - 7444
   - 7447
   - 7474
   - 7477
   - 7744
   - 7747
   - 7774
   - 7777
   There are \(2^4 = 16\) lucky numbers with 4 digits.

Now, we need to find the 17th smallest lucky number. We have:
- 2 lucky numbers with 1 digit
- 4 lucky numbers with 2 digits
- 8 lucky numbers with 3 digits

So far, we have \(2 + 4 + 8 = 14\) lucky numbers. The 17th lucky number will be the 3rd smallest 4-digit lucky number.

5. **Listing the first few 4-digit lucky numbers:**
   - 4444 (15th)
   - 4447 (16th)
   - 4474 (17th)

Thus, the 17th smallest lucky number is 4474.

The final answer is \(\boxed{4474}\).