Three Unique Primes

What is the largest integer less than 100 which cannot be written as the sum of three unique prime numbers? Less than 1000? Less than 1 million?

For example,
10 = 5+3+2, all unique.
11 = 7+2+2 = 5+3+3, but neither sum has all three primes unique.

Extension: The problem above actually has been investigated thoroughly. Try also finding the largest integer in each range which can be written as exactly one such sum.

Source: Original.

Solutions were received from Jeremy Galvagni, Joseph DeVincentis, Philippe Fondanaiche, Denis Borris, Saw L.B., Paul Botham, Carlos Rivera, and Sagar Chivate.

It was quickly pointed out to me that this question is a rewording of Goldbach's Conjecture. The answer in all cases is 17. For the extension, the answer in all cases is 40.
Other sites to investigate:
http://mathworld.wolfram.com/WaringsPrimeNumberConjecture.html
www.utm.edu/research/primes/notes/conjectures

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