Donkey Kong Country

If you can stand listening to Cranky Kong's constant rambling, you'll notice this comment about the Game & Watch series.

Donkey Kong Country 2: Diddy's Kong Quest

Three Nintendo characters appear in the very end of this game. After you've beaten Kaptain K. Rool and listened to Cranky Kong, he will take you to an area where you'll be rated against three other video game heroes - Mario, Yoshi, and Link. If you find over 19 of Cranky's hero coins, you'll take Link's place, over 29, you'll take Yoshi's, etc. Also notice Sonic the Hedgehog's shoes and Earthworm Jim's gun by the no hopers sign.
-
Mario, Yoshi, and Link

Donkey Kong Country 3: Dixie Kong's Double Trouble

A small Mario reference is made here. Sometimes, when you enter Wrinkly Kong's save cave, she will be playing her N64, and a version of the "Inside the Castle Walls" theme will play in the background.

Additionally, a brief Link reference is also made. In Bazaar's general store, after buying the shell, ask him about the castle. If you refuse to pay, he will tell you that someone named Link came in and also asked him about it.
-Link Dialogue 1
-Link Dialogue 2
-Link Dialogue 3

Earthbound

One very minor allusion is made to the arcade version of Donkey Kong in this game. In the Onett video arcade, the arcades have various pictures on their screens. On two of the machines, the first level of Donkey Kong can be seen (first machine to the right of Ness in the back row, second to the left in the front in the screenshot). There are other machines like this throughout the game. Notice the layout of the platforms, and the tiny shapes in the same places as the ladders on Donkey Kong. I think it's safe to assume that the shapes at the bottom-left and top-left are Mario and DK...
-
The Onett Arcade

Kirby Super Star

Boy, there are a lot of them here. Let's see...

First of all, two cameos are hidden in the Stone ability. Stone will turn Kirby into different rocks and statues randomly, and when executed in the air, will smash enemies from above. Very rarely, Kirby will turn into these gold statues of Mario or Samus. These screenshots took the better part of forever to get.
-Mario Statue
-Samus Statue

Mario also appears a second time in Super Star, along with Luigi, Toad, Birdo, and Peach. During the boss battle with King Dedede, and in the Megaton Punch minigame, they can be seen on both sides of the crowd. Most of the time, Peach isn't visible (you can see a small part of her hair and an ear behind Mario in the screenshot below). In rare occasions in the Dedede battle, the screen will shift away after Dedede loses, and briefly reveal her face. This is difficult to do, though, and can only be done in the Arena game, not in Spring Breeze.
-The Mario crowd

Now the fun begins (cracks knuckles). In the fourth sub-game of KSS, Great Cave Offensive, Kirby has to find his way through a giant underground maze. Along the way, there are several treasures to collect. A number of Nintendo references are hidden among the treasures, which can be seen in this inventory screen... Recognize anything? From left to right, starting with the top row, there's the Screw Ball from Metroid, Falcon Helmet from F-Zero, Turtle Shell from Mario Kart, Mr. Saturn from Earthbound, Kong's Barrel from the Donkey Kong games, and last but not least, the Triforce. Those who have played Mario and Wario will also recognize the bucket from that game (third item in the third row).

And to end the chaos, Mario pops up one last time in the ending credits for Milky Way Wishes.
-Mario in the credits

Kirby's Dream Land 3

Another sign that the people at Hal like Metroid. In level 5-2 of Kirby's Dream Land 3, just like every other level, you must accomplish a certain task and obtain a heart crystal. The person who will give you the heart crystal is Samus. At one point in the game, you will enter a cave with three rooms. These rooms contain a total of six Metroids. If you freeze them all with Ice power, then Samus will remove her helmet, and give you the crystal.
-
Metroid on the map screen
-Samus
-Metroid
-Samus without her helmet

Legend of Zelda: A Link to the Past

In certain houses, there are portraits on the walls with a familiar face on them. Though most of his face is shrouded, you can see Mario's hat and eyes looking down at the floor.

Mario Is Missing

Yoshi makes a few more appearances in the Super NES version of this game than he did on the NES. He appears in the opening plot, and on the title screen (interestingly, he's missing his spines). He isn't miniaturized on the map screen like he was on the NES. And he also appears in the cities once you've returned all the artifacts.
-
Title screen
-?
-Map
-With Luigi

Mario Paint

Yoshi makes a few appearances in this game. He pops up on the title screen, for one. There are two Yoshi stamps in the freelance mode (one taken from Super Mario World, another completely original one). One of the pre-programmed portraits is a scene of Mario riding Yoshi with a Mario World background. Also, a baby Yoshi appears in the music mode, and makes the standard Yoshi call.

SimCity

The first (and most easily accessed cameo) in this game is made by Bowser. One of the disasters you can plague your city with is a monster attack. A gigantic Bowser will stomp through your city in search or Mario and Luigi, leveling everything in his path.

If you create a city that reaches a population of 500,000 citizens, you will be awarded with a Mario statue. The land value surrounding the stature will increase once you place it. It's an interesting feature, but a little useless that far into the game.

If you complete all the scenario modes, you will unlock two bonus scenarios. One of these is a town called Freeland, in which there is a forest that has grown into the shape of Mario's head.

Stunt Race FX

In most of the courses, there are billboards along the roads that have pictures of Mario, Fox McCloud, and Kirby on them. On some tracks, you can see smaller pictures of Kirby and Luigi above the stadium.

Super Mario RPG: Legend of the Seven Stars

Seems like Hal isn't the only company who likes to sneak lots of unusual, easily-overlooked cameos into their games. First, when you beat Bowyer, go directly to Rose Town and stay at the inn. The innkeeper will let you stay for free. When you wake up, Link will be in the room, sleeping in the normally empty bed. If you try to talk to him, a short Zelda tune will chime.
-
It's Link

Even more curiously, you'll run into Samus twice in SMRPG. Her first cameo is in Booster Tower. She appears as a toy in Booster's toybox, to the left of the Toadstool doll. Her second cameo is a bit more complicated. After you beat Yaridovich in Seaside Town, head all the way back to the Mushroom Kingdom and visit the castle. She is staying in the guest room, resting up for Mother Brain.
-
Samus in the Toybox
-Samus resting

And finally, there is a small reference to the Star Fox and F-Zero games. At the Hino Mart in the Barrel Volcano, one of Hinopio's shelves is decorated with small models. These models are of an Arwing from Star Fox, and of the Blue Falcon and Fire Stingray from F-Zero X.
-
Model ships

Super NES Super Scope 6

In type A of the Lazerblazer sub-game, Mario will fly across the screen in a small plane, being pursued by Lemmy Koopa on a big missle.

Super Play Action Football

After the team select, a coin will be flipped to decide which player will play offense and which one will play defense. You can choose whether you want heads or tails. Mario's face is on the head of the coin, and a raccoon tail is on the tail side.
-
Coin Toss

Tetris Attack

Originally a girl-oriented game that had nothing to do with Tetris or Yoshi, Nintendo decided that it wouldn't sell outside of Japan. Thus, they reprogrammed the game, adding in Yoshi's Island characters and a few Yoshi-related themes.

Yoshi's Cookie

Just like the NES and Game Boy versions of Yoshi's Cookie, Yoshi is given all the credit for the game even though Mario has a sizeable role in the game. However, Yoshi does pop up during gameplay this time.

Yoshi's Safari

Mario's role in this game is a unique one. He appears on the title screen holding a Super Scope (never to be seen doing so again until Super Smash Bros. Melee), and during the story mode. But during gameplay, he is nowhere to be seen. Supposedly, the player holding the Super Scope is supposed to be Mario. No other Mario-related game has taken this perspective.