Is there a record for threefold repetition for when the claimed positions are the farthest apart, in terms of moves? That’s one long question, so let’s break it down. Threefold repetition, as many know, is when a draw occurs, after the same position, with all piece types on the same square, all possible moves and rights are the same, happens three times in a game of chess. The draw can be claimed whether or not the positions are consecutive or not.

My question is based on the non-consecutive part. Is there any sort of or record for when a draw was claimed for threefold repetition, when the three congruent positions, are the farthest apart by number of moves?

Essentially, the “number of moves” means this: Somtime in the game, position one occurs. So and so many moves later, position two happens. Then the third and final position is reached at last. 

One could view this in two ways: What is the greatest number of moves between positions one and three, OR,  What is the greatest number of moves from position one to position two, and then from position two to position three?

The game must have ended on the basis of a threefold-repetition claim.
I would say 50 moves (possibly 49).

Three-fold repetition requires the game to return same position 3 times.  This obviously can't happen if there are any captures or pawn moves, so therefore as long as the 3-fold repetition is possible so is the 50-move rule - which provides a hard upper limit.