Grinning Totem sez:
,
, Sacrifice Grinning Totem: Search target opponent’s library for a card and remove that card from the game. Then that player shuffles his or her library. Until the beginning of your next upkeep, you may play that card. At the beginning of your next upkeep, if you haven’t played the card, put it into its owner’s graveyard.
So, yes, Totem has a triggered ability, which can be responded to with instants. The ability granted by Totem doesn't end until the trigger resolves.
I believe this is only partially correct and in the end, gives a wrong answer. It is true that Grinning Totem has a triggered ability which can be responded to. However, that ability has nothing to do with playing the card in question.
Grinning Totem also generates a continuous effect when you activate its ability. This affect allows you to play the card "until the beginning of your next upkeep."
So, at the end of your opponent's turn, you activate the totem and place the ability on the stack. Your opponent uses his spell for the turn to Stifle that ability and you use your spell for the turn to counter the stifle.
You both pass priority a few times to let everything resolve and his turn ends. The static ability that would allow you to play the spell is still active.
You move to your untap step and the static ability is active, however nobody receives priority.
You now move to your upkeep. At this point,all 'until the beginning of upkeep' effects end and triggered abilities go on the stack. Once this is done, you receive priority. This is the first time since the totem ability resolved that you had the opportunity to play a spell and its too late.
