When can a losing Lannister player play an Infamous! event? He cannot do that in the "challenge resolution" action window, because only cancel/save responses can be played during that time, right?
Bear with me here because this comment has me a little confused about how you are seeing the basic Response timing in framework action windows. If you know all this, I apologize, but I want to make sure we're starting from the same point.
The steps for resolving the "challenge resolution" framework action window are as follows (all steps together comprise a single timing window):
1. Initiate challenge resolution: The framework events are, in order, a) determine winner, b) settle claim effect if applicable, c) award unopposed power if applicable, and d) award Renown if applicable.
2. Save/Cancel responses: If you have a Response effect that uses the word "save" or "cancel" that legally interacts with the effect initiated in Step #1, you interrupt the initiation and resolution of that effect by playing the Response here. ("Infamous!" does not qualify.)
3. Resolve challenge resolution: You resolve the framework events in order, assuming their resolution is not altered by an interruption from a save or cancel effect. (Technically, you cycle through Steps 1-3 four separate times, once for each framework event)
4. Passive effects: After you resolve the final framework event in its Step #3, you move on to a single (and collective) Step #4 for all of the framework events. That means a passive activated by winning the challenge has to wait until all 4 framework events are resolved - it would not happen after you win the challenge but before claim is settled, for example.
5. Standard Response effects: This is where you may use Response effects that do not use the words "save" or "cancel" to absolutely anything that has happened in Steps 1-4 above (or earlier in Step #5). Again, there is a single, collective Step #5 for everything that has happened. You cannot play a Response to winning the challenge before settling claim, awarding unopposed, etc. This is where, in the "challenge resolution" window, you would play Infamous!.
6. End: When all players have used all the Responses they want, the action window closes. This is also where cards killed, discarded or otherwise removed from play in Steps 1-5 are physically removed from the table.
So technically, the losing Lannister player in your example is playing the Infamous! event in the "challenge resolution" action window. He is just not interrupting anything the way a save or cancel would. You are correct that he could not play the event in Step 2 (it wouldn't mean anything in Step 2, anyway, since the power is technically not claimed until Step 3), but he does play it in Step 5, which is also part of challenge resolution.
Is it possible to play Infamous only once this framework action closes? Does that mean that the Lannister player may move all 4 power just claimed by an opponent to one of his characters? Any help would be appreciated!
Two different questions here:
A) It is "possible" for the Lannister player to play Infamous! more than once in the same challenge resolution (both would be played, one after the other, in Step 5), but it won't do him any good to play it more than once. The first event specifically moves the power claimed from (in this case) the Greyjoy House card to a Lannister character. The second copy of the event would then try to move exactly the same power from exactly the same place. But since the power has already been moved, you can't move it a second time and the effect of the second event "fizzles." I can't move something from your Greyjoy House card that is already gone from it!
B) In your scenario, the Lannister player would only get 2 total power. This is because the 2 power the Greyjoy player won for "claim" has been moved from the Lannister player's House card. It has not technically been "claimed" (regardless of what the "challenge prize" process is named). In game terms, power is "claimed" when the total number of power on cards in play (including the House cards) increased. Power is moved when only the distribution of power changes without changing the total number of power in play. Infamous! can only steal "new" power just taken from the general pool, not "old" power just shifted from one card to another. So the 2 "new" power claimed on the House card for unopposed is the only power Infamous! can steal. It cannot touch the 2 "old" power shifted from the Lannister House card to the Greyjoy House card for the "claim prize" of the power challenge. It also cannot touch the 1 "new" power claimed directly by Balon for Renown.