Capcom Arcade Stadium

After having issues with the other two solutions, I played Dynasty Wars and unlocked it.
Difficulty Level: Hardest
Game Speed: Normal

I hope this works for everybody else. Good luck!

This one is simpler than you might think. Here is what you need to know:

* You CAN use continues (on games that have them). So definitely avoid Pirate Ship Higemaru, Commando, and Vulgus among others because they have no continues.

* You must play in the Normal mode for it to count (no Special, Score, or Time attack modes).

* In the game's option menu, make sure the top option says "Hardest" and has all of the bars lit up next to the word "Hardest"

* Per RAINKINGX1, using fast forward seems to cause this achievement to not pop at the end of the game. Just keep that in mind when doing your playthrough

* Credit to Esu for mentioning this: Using the paid invincibility DLC while doing this WILL NOT get you this achievement.

NOTE: This achievement seems to be very picky. Some people haven't had luck with certain games (including Final Fight mentioned below). However a handful of people on here have said that 1941 works better than most of the other games.

The settings required are as follows:

All settings to hardest
Difficulty Increase Rate: Very Fast
Enemy Fire Speed: Very Fast
Starting Lives: 3

I do not take credit for this. Thanks to everybody in the comments for their feedback. I apologize for any inconveniences.
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Original solution is below:

With those to keep in mind, I chose Final Fight as the game. There are others, but I found this one to be the most straightforward and it took between 40-50 minutes for me to complete on Hardest. First of all, you have unlimited continues to beat it. Secondly, compared to other games in the compilation, you only have to beat the game once.

There are more enemies and they take more hits than on Factory Settings to kill. Otherwise, just burn through the game with whatever character you desire. I personally prefer Cody or Haggar, but again it doesn't matter.

Also, when you lose all of your lives and the continue screen with the dynamite starts to show up with your character, don't spam the buttons used to control the character. I accidentally ran out the continue clock and had to start over from the beginning (I only had to work back to Stage 3, though, no biggie).

Also, don't be afraid to rewind and use save states if needed.

Plus, this will stack with the achievement for completing a game irregardless of difficulty, so you can knock them both out at once.

I hope that helps you out and if you leave a negative vote, please tell me why and I will do my best to improve the solution.

I recommend playing Giga Wing.

Anyone can clear it by using Continue.
Also, you can clear it in about 30 minutes.

My setting
Game Region:Giga Wing JP ver
Difficulty Level:8
Game Speed:Normal
Play mode:Normal mode
Game speed:Normal

I didn't use invincible cheats and rewinds

I did this on Final Fight. Whenever I was on the game over screen with the dynamite I just inserted a coin and continued.