It's not $3$

By the fermat´s little theorem the remainder after division of $2^{p+1}+3^{p+1}+5^{p+1}+7^{p+1}$ by any prime number p is $2^2+3^2+5^2+7^2=87=3*29$ and thus the largest prime $p$ which divides $2^{p+1}+3^{p+1}+5^{p+1}+7^{p+1}$ is 29.