This is a rarely known one: on Football Manager, I once got this gob-smacking message:
"The club has been subject to a takeover and the new board have decide to bestow you with a £500m war-chest. Press 'Accept' to receive your budget."
Holy smokes!?! Hitting Accept as fast as my racing heart and trembling fingers would allow, I was rewarded with the heart-breaking truth: "April Fools!"
Yes, it was an April Fools Day prank, triggering when the computer data and the game date are both April 1. The depths of videogame cruelty knows no bounds.

In Monkey Island 3 you can find the secret other side of the mysterious stump that in Monkey Island 1 requires to:
insert "disk #22," "disk #47", and "disk #114", to continue.

It was pretty cool to find Duke Nukem in the Dark Carnival level in Blood - He was not in the best shape to say the least. He got the same treatment Doomguy got in one of the Duke Nukem3D levels!

Also, the waterfalls in Shadow Warrior had some pretty nice easter egg that was wielding uzis + Tomb Raider.

The one in The Witness where you can see the game secret ending at the very beginning by solving an environmental puzzle with the sun.
You unlock a secret room where as you walk ahead you start seeing a lot of crazy stuff and at the end you wake up in the real world.

The secret moves on Tomb Raider, like the hanstand and the swan dive... With and without water

The Monkey Island scenes, the Barbery Coast in 3-dimensional glory and the possibility to turn Indiana Jones into Guybrush Jones™, wielding a chicken-shooting cannon or was it a gun? in Indiana Jones and the Infernal Machine.

Star Trek Elite Force: II is one of the first FPS I played as a kid. My Father happened to own a decent collection of late 90's and early 2,000's FPS (absent among them Star Trek: Elite Force Voyager).

I remember the first time I played I made a somewhat poor effort to find the many secrets and collectibles hidden on every level. I'm a smart adult, but maybe not so much a smart kid. A guide was definitely used, so points to me for being at least clever enough to think of seeking one.

All of them were gold starships that, at certain thresholds, unlocked some bonus content like wallpapers, artwork from the game's development, a few levels to play in, etc. Some were hidden rather obtusely. One level has you go up and down a space ladder for the gold starship collectible to spawn, and I am still salty about that to this day.

But one of these hidden areas holds a very special easter egg that I retell the story of to anyone who'll listen, even my beloved girlfriend.

It's the fourth mission of the game, on the Attrexian Space Station that's under siege by... well, giant space bugs with impulse engines on their butts. It's a lot cooler than I'm describing it but, that's really besides the point.

Your objective is to escort an Attrexian Engineer through the station and protect him from the giant space bugs with impulse jets on their butts. Yes, I know it sounds immature to call them that rather than their real name. No, I will not stop. Please stay with me.

The Attrexian Engineer stops at a locked door, and you're tasked with zapping the space bugs with your phaser (or something heavier) while he... performs the future equivalent of the janitor fumbling a large row of keys trying to find the right one. We're extremely fortunate that the system doesn't lock you out after a number of failed password entries.

Anyway, after you protect the Attrexian Engineers buns from a space munching, he opens the door for you and behind it is a lot of ammo (you'll be low about now) and a new weapon (it's pretty great. A lightning gun that, as an alternate fire mode, shoots a gas that fills the room and is HIGHLY conductive. Don't be close!)

Furthermore in the room is a false wall. Obliterate it with your phaser and watch it crumble to reveal a long silver tube. Jump down the chute, and enter a Starfleet approved level of Super Mario. You'll find the bulk of the Gold Starship collectibles there.

I hadn't even played Super Mario at the time, and to this day I still haven't. Perhaps I never will.

So then, why does it stand out so much to me in my memory? Why do I retell the story again and again, annoying my friends and loved ones? Why does it even merit a post at all?

Simple.

I thought it was really cool.

In Wolfenstien: The New Order, when you go to sleep in the secret base, you are able to play Castle Wolfenstien which is a game inside a game and those are always amazing.

In Torin's Passage (A sierra adenture game) near the end of the game when you meet the sorceress Lycentia.

You use the bagpipes on her and you get a personal message from Al Lowe!

The easter egg of the Holy Hand Grenade from Monty Python in Fallout 2

I really liked how Ghost 1.0 handles you when you take the instructions too literally when naming your current run. If took the correct naming path, the game closes. Included an image for those who wanted to know. :D

The secret "meet the devs" ending in Chrono Trigger

By far: the "Star Wars body" in Fallout New Vegas

It's subtle and doesn't seem out of place... but if you get it it's one of the best easter egg I ever saw, because it's subtle and not an in your face: "ahah look a funny thing from another art piece"!

The Sarcastic Announcer you can unlock in Wave Rave Blue Storm on the gamecube.

"You have chosen poorly"

"ooooh a tuurboo!"

In Heroine's Quest, finding Hera's ring.

In Quest for Glory 5, the Hero had the opportunity to woo and marry one of four women in the game, proposing by giving her Hera's ring. In Heroine's Quest, you can find it in the Heroine's inventory at the start of the game... IF you name her after one of the four women. (Erana, Elsa, Nawar, or Katrina)

Myself and a friend both had the ring in our first playthrough, figuring it was there for everyone as a simple callback to QFG5, but then his brother didn't have it prompting questions on the community forum where everyone thought we were talking about ANOTHER Easter Egg. (a ring cursed such that, once you take it, it stays with you through ALL playthroughs and across ALL saved games until you finish the quest to get rid of it. On its own, also worthy of inclusion here) The developers, however, did confirm that it was not a bug.