— Burt Kaliski Jr. (@modulomathy) June 23, 2012
Answer: The next tri-composite after 174 is 182, which is the product of the three distinct prime factors 2, 7 and 13. (I found this by trial and error, testing consecutive numbers.)
Note: One might argue that 180 is also a tri-composite, because it is divisible by three distinct primes, 2, 3 and 5. However, it is not equal to their product; both 2 and 3 are repeated factors. The definition of tri-composite given requires that the number be the product of the three prime factors. A different definition might count just the number of distinct prime divisors, in which case 180 would be the next tri-composite.