You are never required to obtain additional destination tickets cards, even if you've completed the ones you have.

You have three options available to you:

* You could claim a route. Remember, routes are worth points, and nothing prevents someone from claiming routes once their destination tickets have been completed.
* You could pick new destination ticket cards. You might get lucky and get one or more that's already completed! If you don't think you're winning, this can be very effective.
* If you're unable or unwilling to take the above actions, you can effectively "pass" by drawing more transportation cards.

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While I used the terminology of *Ticket To Ride: New York*, this answer applies to all of the numerous versions of *Ticket to Ride* I've played (except that one might have some additional actions available, such as building a train station in *Europe*, and buying an upgrade in *London*).