The questioner invited us to ignore illegality or dead position rule.
Title: question: which legal position has the smallest number of moves in a "forced loop"?
Forced draw by repetition in the shortest number of moves. Well that's 4 single moves = 2.0 double moves. Same number of units as Rewan Demontay, but slightly lighter position:
[FEN "5b1k/4p1p1/4P1P1/5pP1/5PpK/6P1/8/8 w - - 0 1"]
Eventual draw by repetition, also ignoring 50/75 move rules Well just two kings on the board is enough. There are 3,612 (= 3655 + 2458 + 4*60) arrangements of the two kings. Each might be achieved with either white or black to play, and appear twice. So after at most 14,448 single moves (=7,224.0 double moves) a position will occur for the third time. Does the connectedness of the position graph allow this value to be achieved "in practice"? Can the draw be postponed this long? I don't know, but I guess yes.