British actor Vic Gallucci holds the record for most appearances as a television extra. Since 1989 he has appeared 819 times as Detective Constable Tom Baker in police TV drama The Bill.
He has yet to speak a line.

What a cool role. I’ll bet 5 whole dollars there’s a fan site out there somewhere for this guy; ideally, documenting all his appearances:

May 13, 1997: Was asked to fetch pastries. Nodded. Returned 13 minutes further into the episode with pastries.

They should shoot a whole episode focusing on Detective Constable Tom Baker as he goes about his duties. Still never give him a line.