Hint: Remember that he can reuse the ends once he's smoked a cigarette.

Answer: 68.

From the initial 274 he can smoke 54 with 4 remaining. From these 58 stubs he can smoke 11 with 3 remaining. From these 14 he can smoke 2 with 4 remaining. From these 6 he can smoke 1 with 1 remaining. Therefore he can smoke 54 + 11 + 2 + 1 = 68.