I remember a lot of these. I've played more old games than new ones, which'll probably show:
Epic Pinball, a pinball simulator for DOS:
I first discovered an extra ball cheat by chance ('twas a single key), but then realized the actual Easter Egg that reveals the cheat (which made the whole deal a lot more satisfying). The game had an inbuilt manual, and after the howto pages, it was basically a catalogue of Epic Megagames. After the "last" page that said something like "End of document! Go enjoy Epic Pinball!" there were still two hidden pages, one being something like "We told you, it's the end of document!" and the other was "ok, you're smart enough, have the cheat!"
apt, a package manager for Debian and many derivative Linux distributions, contains a classic one:
As does aptitude:
aptitude install moo, aptitude install -v moo and so on, I think
Microsoft Office products Word and Excel 1997 contained
a pinball game and a flight sim, respectively.
Commander Keen has these aplenty.
The maps contained lots of texts in The Standard Galactic Alphabet. Cool guys like me could decipher them. The texts were often pretty mundane ("exit", "no humans allowed" or something) but it was fun to crack them anyway.
Most games contained secret levels. My favorite is the one in Commander Keen 4 - it was visible on the main map even if you had no clue on how to get there, and I spent ages pondering how to get there, eventually succeeding - by chance. 12 inches = 1 foot.
Command and Conquer games feature these as well.
Both C&C1 and Red Alert 1 featured secret campaigns and Tiberian Sun has a "secret" mission (scare quotes because I accidentally found it on my first playthrough, so not really that secret).
Who could forget the secret maps of Doom II?
They were based on Wolfenstein 3D, and tried to emulate the considerably less advanced 3D feel it had. They contained SS officers ripped straight from Wolf 3D, and most wall textures were also taken from Wolfenstein. One of the levels even featured a secret area which downright mocked the Wolf 3D by containing all stuff Doom had but Wolf didn't - variable elevation, lightning effects and such. The second secret level is finished by killing hanged Commander Keens, by the way.
Speaking of Wolfenstein 3D:
One of the original Wolf 3D episodes was supposed to have a level with a hidden area containing instructions to participate in a contest. It was aborted as the first map editors made the secret area visible sooner than they thought.
BioMenace, a platformer based on the Commander Keen engine, had a nice one:
You could actually meet the DevTeam. And you could try to kill them and wind up in a casket yourself.
ADoM, like most Roguelikes, has many little details left for the player to find. This one isn't encountered during normal gameplay, and is of no advantage, but:
Try using the creature description tool on "Thomas Biskup". Some of his friends are also included, I think.
And just to get a console game on the list, some Ratchet and Clank titles have
the Insomniac museum, typically accessible during night only.
Since I could go on forever, I think I'll leave this here.