Donkey Kong Jr.

The sequel to DK marks the first, and last time Mario was ever a villain rather than the hero. He stands at the top of each level, sending things to attack DKjr, such as Klaptraps, buzzards, electrical currents, and crows. When you beat the game, Mario falls to the ground, using the exact same sprite that was used when you fell too far in DK.
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Mario the bad guy
-Mario defeated

Donkey Kong Jr. Math

This can definately be considered one of the first educational titles. Likely to help sell the game, Nintendo themed it around the Donkey Kong games. But for some reason, DKjr gets all the publicity. To make it more interesting to younger kids, possibly? However, Donkey Kong appears in the various play modes. In the A and B challenge games, he will hold up a number, and you and your opponent must race eachother to gather numbers which will add up (or subtract, divide, or multilply) to the one DK has.
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DK Jr. Math, A/B mode

DK does something slightly different in the third game mode. First, you must select which kind of problem you want to solve, as DK indicates with his sign. Then, while you solve the problem, his ever-present sign will display your score. What's with all these signs, anyway?
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Select your excercise
-Yet Another Sign

Earthbound Zero

At one point, you'll run into a kid who asks if you've played Super Mario Bros. 7, and then comments on SMB3. Thanks to ZNess for this screenshot.
-Have you played Super Mario Bros. 7?

Golf

One of the first Nintendo sports titles, and Mario's first golfing appearance. Basically he just stands in the lower left corner and only moves during strokes. That's about it.
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Stroke Play
-Putting

Kid Icarus

An interesting reference to Metroid is made in it's companion game, Kid Icarus. In the final vertically-scrolling level, Metroid-like enemies called Komaytos will attack pit in groups of four. The instruction booklet even makes reference to them coming from another planet. Thanks to The Metroid Database for this screenshot and info.
-Komaytos = Metroids?

Maniac Mansion

Maniac Mansion is a strange game, but just the right kind for a brief Mario reference. To enter the last room in the Mansion, you have to activate the arcade game Meteor Mess. When the high score screen appears, it will give you the four-digit code to the security door. Interestingly, when the high score screen pops up, the first five notes of the Super Mario Bros. theme play.
-Meteor Mess
-High Scores

Mario Is Missing

Yoshi plays a small role in this game. In fact, the game would have been virtually the same without him. But Yoshi was really popular at the time, so... Yoshi waits in Antarctica while Luigi returns the stolen artifacts. Then, once all the artifacts in a city have been returned, you have to lead Yoshi to the city Luigi is in, and jump into the nearest warp pipe to get back to Antarctica.
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Luigi and Yoshi
-Yoshi on the map screen

Mario's Time Machine

Yoshi's role in this game was also minor. He appears at the beginning, before entering the museum, and at the end after rescuing him from Bowser. He also pops up as a portrait in the hallway, in the newspaper, and during the ending credits.

Interestingly, Donkey Kong Jr. also makes a brief appearance in the hallway.

Mike Tyson's Punch-Out!!/Punch-Out!!

The original boxing game, often emulated, but never duplicated. Though he made a small cameo in the two arcade versions of the game in the pre-NES days, he is considerably more noticeable here. That's right, Mario is the referee. He starts the match, and counts down to the knock outs. This also the first time Mario ever had a voice. A bit fuzzy, but not bad for 1987. Also, one of the boxers, Don Flamenco, makes a comment about Mario's hair.
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TKO

NES Open Tournament Golf

Another Mario-related golf game in which Mario isn't named in the title. This time, Luigi is also playable (as player 2), and Peach and Daisy show up as caddies. Toad also pops up here and there. Most interestingly though, is that Donkey Kong runs the prize money booth.
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Mario, you're broke!

Pinball

This was one of the first NES games, and also one of Mario's earliest (and strangest) cameos. On the lower half of the Pinball table, there is a hole in the upper-right corner which leads to a bonus level. In the bonus level, you control Mario, and have to bounce the Pinball on the bumper he is carrying, while Princess Toadstool runs back and forth on a platform above. Once Mario has bounced the Pinball against the upper platform, highlighting the numbers in the center of the screen enough times, the platform will disappear and Toadstool will fall to the ground. You must then catch her on Mario's bumper for a bonus, or she will become a flattened mess.
-The Minigame

Startropics 2: Zoda's Revenge

Being such a unique game, it's no surprise that ST2 contains a silly Mario reference. When you reach Egypt, Cleopatra will ask you to go to Rome and get a pizza for her. After you exit a dangerous cave, and borrow a camel, you'll walk...and walk...and walk...and walk...and walk...for a very long time. Then, you'll run into a man riding on the back of a koopa troopa, who will give you Cleopatra's pizza. Interesting that he's native to Italy...
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You've got the Pizza

Tennis

An early Mario sports appearance. This time, Mario referees the game, and makes various calls, like in, out, fault, double fault, and so on.

Tetris

If you beat game B on level nine, Nintendo characters will appear on your winning screen, playing different instruments. Which characters and how many appear at once depend on what height you beat the level at. At the lowest height, Pit will appear, playing a violin. At next, Link will join him playing a flute. Then Samus playing a bass, DK with a drum, and finally Princess Toadstool, Bowser (with an accordian), and the Mario brothers doing a funky Russian dance. Nice touch. Thanks to The Mushroom Kingdom for this screenshot.
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High Score Screen

Wrecking Crew

Another very early cameo, released around the same time as Golf, Tennis, and Pinball, although a bit more unique. Wrecking Crew hands Mario a sledge hammer and puts him in one construction (or is it destruction?) zone after another. Luigi also appears in the two-player mode.
-Mario's ready
-Mario start!

Yoshi

In the years following Super Mario World, Yoshi was rather popular and appeared in several title games, even though Yoshi himself was rarely playable. In this game, there are a few Yoshi references (the Yoshi eggs), but the game is mostly Mario-related. In fact, you control Mario during the gameplay. He also makes a few other appearances.
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Gameplay
-Congratulations

Yoshi's Cookie

This is even more inexplicable than Yoshi. Again, Mario appears in most of the game, while Yoshi only pops up here and there (e.g. Yoshi-shaped cookies), yet Yoshi gets the credit. Interestingly, the cookies in this game were later used in the modernized version of Vermin in Game & Watch Gallery 2, and Egg in Game & Watch Gallery 3.
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Gameplay
-Select your options