It's already been said, but I have to second the Elder Scrolls games. The Demon/Dark Souls games are a match too, but when you talk about:

* No fixed, linear story that you must complete, and

* Hidden secrets,

you are really talking about a traditional Western RPG, and the Elder Scrolls series is the epitome.

When you finish the opening sequence and you are dropped into a gigantic open world -- yours to explore and make an imprint on as you see fit -- I think you will realize you found what you were looking for.

--- edit ---

Dark Souls fits those requirements too, but they are also unlike most games in the genre in that they are brutally difficult and taxing in an action-oriented sense. If you want edge-of-your-seat combat, that's definitely the road you want to go down, but you do sacrifice some in the sandbox and exploration department.