I'm not sure if this is quite what you're asking for (it seems like you're looking for extras that are in some way weird or unusual things to have as an extra), but there are a few games on GOG where the included materials (manuals, reference guides, inserts, etc) really go above and beyond the call of duty in acting as an extension of the game world and immersing you in the theme. A few examples include:
-Arcanum - Of Steamworks and Magick Obscura: probably the greatest game manual ever, it has extended (and entertaining) essays on the game universe and its races delivered as a series of antiquated science lectures. If you love the game but haven't checked out the manual, you're missing out.
-Wing Commander 1 + 2: Many of the reference materials are written as newsletters and manuals that would be handed out to actual crew members aboard the ship. (Later Wing Commander games may continue the trend, but I wouldn't know since I haven't gotten to them yet.)
-Crusader - No Remorse: Similarly, many of the equipment guides and other materials are presented as intel stolen from the Consortium's databases by hackers, and contain numerous margin-notes from the in-game characters.
-The Fallout games (may the rest in peace) - all the little PipBoy cartoons in the manuals do a great job of establishing the games' discordant blend of 1950's chipperness with post-apocalyptic misery.
-I Have No Mouth And I Must Scream comes with a copy of the short story the game is based on.
-"Divine Divinity" includes a full prequel novella that sets up the game's events. (Unfortunately, it's quite terrible.)
There are also some games that I consider worth it for the soundtrack alone, including Thunderscape, Entomorph, and Chaos Overlords.
Finally, Roman5 has a GOGMix of games that come packed with tons of extras:
http://www.gog.com/mix/incredible_value_gogs_with_a_huge_list_of_extras