Battletoads in Battlemaniacs had continuous scrolling bonus stages following the second and fifth levels. The object was to collect the bowling pins or dominoes for points and avoid the black ones and enemies.
The BattletoadsArcade Game had a Street Fighter II-like timed stage after the third level to demolish a Rat Fighter like the one from Battletoads and Double Dragon.
Jackie Chan's Action Kung Fu has hidden bells in many places. When Jackie picks up a bell, he is instantly transported to a "special stage," where he can win points, by punching/kicking things or jumping on clouds, that can be spent on health, continues, and Psycho Waves. There's a show of fireworks for a perfect score.
In between certain stages of Dynamite Dux, Bin must fight Pin in a boxing ring for points. If only one person is playing, Bin fights a computer-controlled Pin; if two are playing, they must duel.
Jaws has a bonus stage involving bombing jellyfish from a plane.
Batman Doom, unusually for a vanilla Doom mod, has one of these. During your first boss fight with the Scarecrow, mayor Kroll is tied to a ticking bomb in the vicinity. If the bomb explodes, the level ends and you go to the next one (map16) like normal. But if you manage to actually defeat Scarecrow before the explosion, you go to map31. It's a big, empty (and thus somewhat creepy) city map where you're trying to collect a bunch of random bonuses before you can access the exit and continue onto map16. This is also where you can access the Secret Level.
Devil World has the "Bonus Box" at the end of every round, where Tamagon searches for a 1-Up hidden in one of six boxes. As the Devil has flown away, Tamagon can for once control the direction of scrolling by stepping on arrows.
Mario Bros. had the enemy-free "Test Your Skill" stages, where the objective was to collect coins within the time limit.
In the original Super Mario Bros., going down certain pipes or climbing up hidden vines would take you to hidden areas filled with coins. Later games in the series kept these in-level bonus areas but added entirely separate levels, sometimes hidden, sometimes not, in which you could get loads of points, lives, or items.
Super Mario Bros. 3 has an even larger number of different kinds of bonus area/level. Underground coin rooms, beanstalk coin rooms, the slot-machine Mini-Game, the Memory Mini Game, the treasure ship, etc.
Super Mario World has, in addition to coin rooms, a Mini-Game where Mario must hit various cycling blocks and try to get 3 shapes in a row to win 1-Ups. It is accessed by accumulating 100 stars via breaking the tape at the end of the main levels (they're absent in Ghost Houses, Fortresses and Castles).
In Super Mario Land on the Game Boy, in addition to coin rooms, taking the harder-to-reach exit to a stage would lead to a bonus game where, depending on your timing, you could get lives or a power-up. One room, notably, exaggerates this trope by having all of the space in the room except for the floor and exit pipe filled completely with coins. It was impossible to collect all coins unless you used a thrown Super Ball (which, unlike other Mario games' fireballs, can collect coins, bounce at a 45-degree angle upon hitting a surface, and are unaffected by gravity). A few of the coin rooms have Spikes Of Doom, allowing you to die in a coin room.
Super Mario Galaxy 2 has scattered Warp Pipes that lead to various rooms with free stuff available. Some contained dice which can give coins, Star Bits, or OneUps. Others contain many coins that, if collected soon enough, give a 1Up.
In Ristar, each level contains a secret entrance to a bonus stage where you must complete a task within a time limit to get a special item; collecting enough of these would unlock passwords that you could enter to modify various aspects of gameplay.
In the first 16-bit Sonic the Hedgehog game, collecting enough rings during a level would unlock a bonus stage (Special Stages) at the end of the level where you could try to win a Chaos Emerald. In later 16-bit era games, these areas would become accessible during the levels rather than after them, and would later be accompanied by additional types of bonus stage (called oddly enough, Bonus Stages) that didn't contain Chaos Emeralds but did contain power-ups and the like, but these were still treated as separate stages rather than as parts of the stage you came from; your score would be tallied at the end of them just like at the end of any other stage.
In the 8-bit (Master System and Game Gear) game, the Chaos Emeralds were hidden in the levels but collecting enough rings gave you access to the bonus stage full of rings, extra lives and one continue per stage.
Sonic 3 was unique for having bonus stages dedicated to the Chaos Emeralds (3D mazes where you must "Get Blue Spheres"), as well as three less elaborate bonus stages for the express purpose of farming rings, lives, and shields — the Gumball Machine, sort-of-Pinball, and Slots.
The bonus stages were brought back for Sonic Heroes, which was notable because the other 3D games (both before and after Heroes) just automatically gave you the Chaos Emeralds over the course of the main plot. Of course, those Emeralds were still necessary to reach the final story...
SonicRush did the same thing, but only for Sonic's story - Blaze got a Sol Emerald automatically after every boss fight. The Chaos Emeralds AND the Sol Emeralds were both necessary to access the final stage in this case.
As sort of a weird example, in Earthworm Jim, each regular level would be followed by "Andy Asteroids?", a race with Psy-Crow. Beating him would allow you to go on to the next stage without incident and possibly rack up an extra continue, while losing to him forced you to fight him before going on to the next stage.
In Earthworm Jim 2, the same was done with a minigame where you had to use a giant marshmallow to bounce Peter Puppy's puppies to safety after Psy-Crow throws them out of a window. If you drop too many, Peter transforms into his Superpowered Evil Side and attacks you.
In Skullmonkeys, the bonus rooms are accessed by collecting the "Swirly Q"s and entering a special exit at the end of the level. It's most remembered for the song that plays there, which must be heard to be believed.
Donkey Kong Country and its sequels have loads of secret areas filled with goodies, some of which behave more like part of a normal level and some of which behave more like levels in their own right.
Donkey Kong Country Returns has hidden rooms that contain many bananas, banana coins, and balloons. Collecting everything reveals a Puzzle Piece necessary for 100% Completion. Falling off does not result in death.
In Cool Spot, collecting 75 Cool Points in a level unlocked a bonus stage after the level was finished in which you could get 1-ups and continues.
In Dynamite Headdy, putting on the Liberty Head transports you to a bonus game where you try to shoot a certain number of basketballs through moving hoops within a time limit. Each time you successfully complete it, you get one of the characters in the four-character password needed to unlock the Bonus Boss after the end credits. You are transported back to the same point in the main level when the bonus game ends.
The first Crash Bandicoot has bonus stages that are gotten to by collecting three of certain items. The later games have one on each level, which you just have to step on a pad to get to. Some levels have a second pad that only appears if you get that far without dying, or have the corresponding colored Gem in your hands, which leads to a more-dangerous-than-normal-play Death Stage.
Jazz Jackrabbit had a variation where, after each world, you would get a bonus round, where Jazz would run around (in Third-Person Shooter mode, minus the shooter part, rather than Side Scroller mode) in a semi-3D maze collecting gems. If you met the target within the time limit, you got a 1-up.
Every world in Purple has one bonus stage which can be unlocked by finding a hidden switch. These stages contain lots of 1-ups.
Bug! has two kinds. The first plays like the regular game itself (except you had to collect gold objects for extra lives). You could die in those, if that happened you'd exit the bonus level. The other is a Pass Through the Rings bonus, with an extra continue should you make it to the end.
Aladdin (Virgin Games) has special tokens you can collect to earn a chance to play as Abu in Agrabah or the Cave of Wonders.
In Aladdin (Capcom), the "A Whole New World" stage is a bonus level with no enemies.
The Lion King has bugs that can be collected to play bonus games as Timon or Pumbaa.
In Nuts & Milk, the third of every set of five levels was a timed bonus round. Yogurt would be waiting on a box in the middle of the screen instead of in a house at the top left, Nuts was replaced by fireballs that moved slowly around the screen, and collecting fruits were optional, but they otherwise played like ordinary levels. However, dying in a bonus round wouldn't affect the number of lives.
Getting more of a certain item than the opponent does, however: that's a Shout-Out to the original game which has a bonus game - In the first Bubble Bobble, the player(s) race to get the most of a type of item on the screen in a time limit.
Mighty Bomb Jack has the unusual inversion of this trope, a Penalty Stage. Players who get too greedy collecting Mighty Coins or Mighty Drinks will be sent to a torture room, which has enemies but no prizes or exits. Escape is obtained by jumping fifty times.
In The Adventures of Lomax, when you defeat 50 enemies, the exit at the end of the current level will bring you to a bonus stage, where you have a limited time to collect loads of money. The stage itself is entirely made out of money, too. No enemies to be found, and dying only brings you to the next level.
Milons Secret Castle has hidden music boxes that transport Milon to an area in which he collects musical notes to earn money.
The credit rolls of the Tetris The Grand Master series have you continue playing. Some modes have you simply continue playing as the credits roll, with no effect on your score or grade. Some others, like TGM2's and TGM3's Master mode, have the "disappearing roll," in which pieces vanish 5 seconds after locking down; in TGM3 clearing lines during this part will add a small fraction of a whole grade. Then there's the infamous "invisible roll" in which pieces vanish upon locking; in TGM2, this is required to earn the Grand Master rank (failing will net you an M grade instead), and in TGM3, this nets even more grades per line clear, and clearing enough lines and surviving are just part of the requirements for TGM3's Grand Master rank.note And believe me, those two requirements alone are hard enough. How hard? Well, out of the millions who've played Tetris, only 3 of them have this rank. Finally, TGM3's Shirase credit roll has you playing with fully-visible double-sized pieces, but has no effect on your grade.
You access these in Angry Birds by hitting the golden eggs in some levels.
In Binary Land, the third stage out of every five was a bonus level where the goal was to collect hearts.
Destroy The Godmodder has had a few. Team Fortress 2 mode in the second game, as well as the first intermission.
The US manual for Deathsmilesadvertises the Ice Palace in MBL as this. Under normal conditions, it is exceptionally easy to rack up points and max out your lifebar (scores of over 11 billion from the stage by itself are not unheard of, making it the single high-scoring stage). Unfortunately, if you happen to be doing toogood, the background will be red instead of blue, and you'll have to contend with it at Rank 999, where accumulating points is much more difficult.
Space Invaders Extreme has bonus rounds that are activated by shooting red or flashing UFOs. Inside, you have to shoot a certain number of armoured or evasive invaders within a time limit. Your reward? When you exit, your cannon becomes a Wave Motion Gun for at least 15 seconds.
Space Invaders Extreme 2 streamlines this feature by having the main stage keep going while the bonus stage takes place on the top of the screen, which you have to shoot upwards to.