[BadgerDog]

Being a basic member myself, you can participate in some of the Black Pearl chapter quest. The game however, requires you to upgrade to unlimited access in order to talk to Gibbs and finish the Carver quest.

You can also take the Tortuga totem quest, but the reward is of no meaning if you don't have unlimited access.

And when you're done with those, ask around with townsfolk, most of them have quests for basic access.

It won't be long though, that you'll end up repeating townsfolk quests. It's Disney's way to make you realize why basic access is free.