score:11
It's the medal of the American Legion:
According to the notes that accompany the picture on the AP website, the photograph was taken:
... after President Johnson's address to the national convention of the American Legion Tuesday afternoon,September 10, 1968.
JBJ was famously a member of the American Legion, and was awarded the Legionβs Distinguished Service Medal in 1968.
Similar lapel badges are still available to members of the American Legion.
The silver star lapel pin:
can be seen above the Legion medal on LBJ's lapel