Find the largest number n positions away from n

05AB1E, 21 bytes

vyN+Ny-})¹gL<Ãv¹yè})Z

Explanation

v      }               # for each num in input
 yN+                   # push index + num
    Ny-                # push index - num
        )              # wrap stack in a list
         ¹gL<Ã         # remove indices outside the range of input
              v¹yè})   # get list of elements in input at the remaining indices
                    Z  # get max

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Haskell, 61 57 55 bytes

f x=maximum[a|(n,a)<-x,(i,b)<-x,b==abs(n-i)]
f.zip[0..]

Usage example: (f.zip[0..]) [5,28,14,5,6,3,4,7] -> 14.

(More or less) a direct implementation of the definition: for each index n of the input list x keep a := x!!n if there's an index i where b := x!!i equals abs(n-i). Find the maximum.

Edit: @xnor saved two bytes. Thanks!

Jelly, 10 bytes

,N+JFfJị¹Ṁ

Try it online! or verify all test cases.

How it works

,N+JFfJị¹Ṁ  Main link. Argument: A (array)

,N          Pair A with -A (element-wise negative).
   J        Yield the indices of A [1, ..., len(A)].
  +         Add the elements of A (and their negatives) with the corr. indices.
    F       Flatten the resulting 2D array.
     fJ     Filter indices; remove invalid indices (not in [1, ..., len(A)]) from
            the generated array. The result is the list of all indices of eligible
            elements of A.
       ị¹   Retrieve the corresponding elements of A.
         Ṁ  Take the maximum.