Normally the tiger would get one AoO on the character; however, the tiger can't make AoO at all while grappling.
From grapple rules:
[..] If successful, both you and the target gain the grappled condition. If you successfully grapple a creature that is not adjacent to you, move that creature to an adjacent open space (if no space is available, your grapple fails). Although both creatures have the grappled condition, you can, as the creature that initiated the grapple, release the grapple as a free action, removing the condition from both you and the target. [...]
And the grappled condition states:
Grappled creatures cannot make attacks of opportunity.
So no AoO in this case. The only way the tiger can make AoO and grappling the character is using this special rule of the Grab ability to prevent itself from taking the grappled condition:
[...] The creature has the option to conduct the grapple normally, or simply use the part of its body it used in the grab to hold the opponent. If it chooses to do the latter, it takes a –20 penalty on its CMB check to make and maintain the grapple, but does not gain the grappled condition itself.
To be more specific and clarify Hey I Can Chan's comment the rake ability gives you 2 free extra attacks...but they are still 2 attacks. You have only one AoO so you can use only one of your attacks. Before the grapple you had only 3 attacks; now you have 5 against the grappled target but, since this is an AoO, you must choose only one of this five to make the AoO.
Making an Attack of Opportunity
An attack of opportunity is a single melee attack, and most characters can only make one per round. You don’t have to make an attack of opportunity if you don’t want to. You make your attack of opportunity at your normal attack bonus, even if you’ve already attacked in the round.
(Emphasis added to quotes)