What is the smallest possible product of one one-digit prime and two distinct two-digit primes?
The smallest possible product is formed by the smallest one-digit prime and the two smallest two-digit primes.  The smallest one-digit prime is 2, and the two smallest two-digit primes are 11 and 13.  Their product is $2 \cdot 11 \cdot 13 = \boxed{286}$.