This question was prompted by a statement in this answer.
Charging Ends Your Turn
After a charge, your turn is over, except for free actions. From the Rules Compendium, pg 240:
No Further Actions: The creature can't take any further actions during this turn, except free actions.
A creature's turn usually ends after it charges. However, it can extend its turn by taking certain free actions.
Action Points are Free
Using an action point to get another action is a free action. From the Rules Compendium, pg 67:
Action: Free action. A creature must have an action point in order to take this action.
Gain an Extra Action: The creature gains an extra action to use during its turn. The action can be a standard, a minor, or a move.
The action 'Spend an Action Point' is a free action, so clearly you can do this after a charge. However, the action you get from using that free action is explicitly not free. It is either a "standard, a minor, or a move".
Therefore, it seems to be pretty clearly laid out that although you can use an action point after the charge, you won't actually benefit from doing so because you can't use the action you gain.
It was suggested that the phrase "it can extend its turn by taking certain free actions" covers this, but I'm reading that as extending the turn because there are more actions occurring during the turn, all of which are free actions.
I suppose you could get other mechanical benefits from spending an Action Point, like the ones on paragon paths or from a Warlord's presence, but it doesn't appear that you could get the extra action.
Am I missing something? What? Where?