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NestWhile[Rest@RotateLeft@# &, Range@10, Length@# > 1 &]

{5}

FixedPoint[If[Length@# > 1, Rest@RotateLeft[#], #] &, Range@10]

Edit

Historical note: As far as I can remember, Josephus roulette (a plain treason to his companions) consisted in killing every third person.

FixedPoint[If[Length@# != 1, Rest@RotateLeft[#, 2], #] &, Range@10]

{4}

Note: The direction is important. RotateRight[] will select another victim.