So you are one of these: 21,22,24,25,26,27,28

The sum of my prime factors is the greatest of all the 
composites between 20 and 30.

So the sum of your prime factors is 28, for that's the 
greatest composite between 20 and 30.

21 has prime factors 3,7, which have sum 10
22 has prime factors 2,11, which have sum 13
24 has prime factors 2,3, which have sum 5
25 has prime factor 5
26 has prime factors 2 and 13, which have sum 15
27 has prime factor 3
28 has prime factors 2 and 7, which have sum 9

But none of them have sum of prime factors = 28.

So there is no solution, unless there was a typo.

Edwin