All prime numbers except for 2 and 3 are either of the form 6n+1 or 6n-1.

A number of the form 6n+1 can be expressed as 9+25+3(2n-11). The first two numbers are obviously composite. The last is composite whenever 2n-11 is at least 3, so when n is 7 or more. So 43, 49, 55, 61, etc. The highest prime number in this class below 43 is 37.

A number of the form 6n-1 can be expressed as 9+35+3(2n-15). Again the first two numbers are obviously composite. The last is composite whenever 2n-15 is at least 3, so when n is 9 or more. So 53, 59, 65, 71, etc. The highest prime number in this class below 53 is 47.

Therefore the answer is 47.

For completeness, the full list of prime numbers that cannot be expressed as the sum of three positive composite odd numbers is:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 47.