GULYADKIN: As you place your force on the map, you may place in your starting area a Gulyadkin the Lion bystander token as described on the back of this card. As long as a friendly Alyosha Kraven is on the map, Gulyadkin the Lion can be given one action as a free action.

Just wondering since this doesn't say once per turn, can Gulyadkin take as many free actions as he wants just as long as he doesn't repeat any, obviously?

Nope, just one.

Look in the PG, on the first page under "Rulebook", it says that powers on a character's card can only be done once unless it has an exception. Ex: WXM Wolverine has a trait that says "At the beginning of your turn, heal Wolverine of 1 damage". You may heal him of one and only one damage using this trait.

The reason why I bring this up is because of this:
MINION: MINDLESS PAWN: Once per turn, if Doppelganger has exactly one action token, you may give a power action to a friendly character with a higher point value within 8 squares. You may now give Doppelganger one action as a free action.

WoS Doppelganger sets a precedent with the last sentence being that the Mindless Pawn power uses the same language. But, it also says Once Per Turn, whereas Gulyadkin does not. This leads me to believe that the lion could take as many free actions it can without repeating one, respectively.

Unfortunately, Wizkids is far from consistent in their use of language. There are several examples out there in which powers that do exactly the same thing are worded differently.

Once per turn, or once per game?

per turn. since the caveat usually says "at the beginning of your turn" or "during your turn", etc.