Answer: 38

Small Hint:

Test candidate even integers by listing the odd composites no greater than half the candidate

Big Hint:

For sufficiently large even $N,$ work modulo $6$ and try subtracting $9,$ $25,$ or $35$

Solution:

In a representation of $38,$ the smaller odd composite would be at most $19.$ The only possibilities are $9$ and $15,$ whose complements $29$ and $23$ are prime. Thus $38$ is not representable.

Now let $N>38$ be even. If $N\equiv0\pmod6,$ write $N=9+(N-9).$ If $N\equiv2\pmod6,$ write $N=35+(N-35).$ If $N\equiv4\pmod6,$ write $N=25+(N-25).$ In each case the second summand is an odd multiple of $3$ greater than $3,$ hence is composite; the fixed first summand is also odd and composite. Therefore every even integer greater than $38$ is representable, so the largest exception is $38.$