If you can't use negative numbers, then it's 0.
If you can use negative numbers, then there is no smallest one digit number.
You're used to counting in a base 10 number system. That's counting where you have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. There are other counting systems. Computers use a binary (base 2) counting system, where you only have the digits 0 and 1. There's also hexadecimal (base 16), where you have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. There's nothing to stop you from using a different counting system. You could use a base 17 counting system, a base 18 counting system, and so on. You can keep going forever, approaching (but never reaching) base-infinity.
In base 10, the smallest single digit (with negative numbers) is -9. In each base n counting system the smallest digit (with negative numbers) is -(n-1). Since n can approach (but never reach) infinity, there is no upper limit to n, so there is no lower limit to -(n-1). Under these rules (negatives allowed, no set base) there is no smallest one digit number because whatever value for n you choose, you could always make a smaller one by then choosing n+1.
If you can negative numbers and you have to stick with a base 10 counting system (the normal way of counting you're used to), then it would be -9.