game world conspiracy

Not sure what exactly you're looking for, but the original Deus Ex game had a ton of stuff about various conspiracies (and was the first time I heard of the Illuminati, before it became some kind of meme online)

Deux Ex is an amazing game - PLAY IT NOW!

Deus Ex, because it deserves to be mentioned three times.
Receiver is also good, but keep in mind that the entire plot is contained in audio recordings that don't have subtitles. Also, it's less "Illuminati" and more like "evil creatures from another dimension".

Where's Carlest? Where's that damned frog when we need him?

For point and click adventures (available for both pc and mobile) I suggest Broken Sword: Shadow of the Templars

Deus Ex series.Can't think of a better example right now.
The whole backstory is based on the premise that the Illuminati are real, and he outcome depends on where in the timeline their influence is enough to make a difference.

Ignoring that, Resonance comes close but isn't quite there. The larger conspiracy is hinted at, but doesn't come up full force. Still very worth playing though, and one of my favorite point & click adventures from the past 2 decades.

The Moment of Silence is a pretty good point and click conspiracy adventure.

Like everyone has been mentioning, I'd say Deus Ex.
However, I'd also like to recommend the Assassin's Creed franchise.

Deus Ex (Please, do yourself a favor and only play 1 and HR, skip Invisible War), Assassin's Creed, Mirror's Edge, Max Payne, and Sleeping Dogs. The last two are more organized crime than conspiracies, but they still apply.

Alpha Protocol :)

Minecraft. Well, to me that is,
I would say deus ex and AC but those were mentioned too many times already

The Secret World is all about conspiracies.

Assassin's Creed (the oldest games more than anything)