Minimum number that cannot be formed by any subset of an array
You can pre compute an array $B$ such that $B[0]=0$ and $B[i]=B[i-1]+A[i-1]$. That you can use $A[L,R]=B[R+1]-B[L]$ to speed up you computations.

Now you can enumerate all pairs $A[L,R]$ and insert them into some sorted data structure $T$, and then you will be able to enumerate all of them in a sorted fashion, and thus answer your question.

This will still be in complexity $O(n^2\log n)$ where $n$ is the size of $A$ but it will not change even if the numbers in $A$ are large. This is not fundamentally faster than proposed solution, I don't know if there is indeed a faster one.