Let $d = \gcd(n + 7, 2n + 1)$, so $d$ divides both $n + 7$ and $2n + 1$.  Then $d$ divides $2(n + 7) - (2n + 1) = 13$, so $d$ is at most 13.

If $n = 6$, then $\gcd(n + 7, 2n + 1) = \gcd(13,13) = 13$, which shows that the value of 13 is attainable.  Therefore, the greatest possible value of $\gcd(n + 7, 2n + 1)$ is $\boxed{13}$.
Question:
Let $n$ be a positive integer.  What is the greatest possible value of $\gcd(n + 7, 2n + 1)$?
Answer:
Let's think step by step.

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13
Let $n$ be a positive integer.  What is the greatest possible value of $\gcd(n + 7, 2n + 1)$?
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