The Disney Afternoon Collection

There is a bug that lets you get this achievement and use the rewind feature.
Steps:
Play any game.
Get to the final boss, I recommend you save here in case you do the glitch wrong.
When you think you can handle the rest of the fight without rewinding, save.
Quit The Disney Afternoon Collection.
Now reboot The Disney Afternoon Collection.
Now load your save and defeat the boss without rewinding.
The achievement will unlock and you should also be given the achievement for completing the game. That is, if you don't have it already.

You can get this by finishing any of the six games and making sure you don't use rewind. I found it easiest to play Duck Tales on easy, but the following additional steps were key to my success. Although I refer to Duck Tales, these tips will work for any of the games.

1. Abuse the save system. You can save your progress at any time by pressing and choosing "Save Game". As you progress through a level, be sure to save frequently so you don't have to replay an entire level. Be especially sure to save at the beginning of any tough parts and at the beginning of the boss battles. If you die or lose more health than you'd like, simply press , then select "Load Game" to resume from where you last saved. Make sure you also save after finishing each level so you don't have to repeat anything.

Note: You only get a single save, so be careful that you only overwrite your save when you've made decent progress and you still have a good amount of health. If you save at a bad time, you could get stuck and have to start over.

2. Use the Xbox Accessories app to map the rewind button ( by default) to something else, such as down on the d-pad. This will prevent you from rewinding by mistake. I used rewind freely in my first playthrough, so remapping it prevented me from screwing up my no-rewinds playthrough on the occasions I hit rewind without thinking about it.

3. If you're not intimately familiar with the levels, make use of maps on the Internet to help you take the shortest path possible to finish the levels. This can help you avoid backtracking and unnecessary reloads. I found these Duck Tales maps quite helpful for this purpose. (Please note that I am NOT the creator of these maps - all credit for these goes to the author(s) listed on the linked page).

Good luck!

Basically for this achievement, you need to play and beat one of the six games included in the Disney Afternoon Collection WITHOUT pressing at all. It's hard, I can understand, but just keep your fingers away from at all times.

DO NOT ATTEMPT THIS ON TALESPIN! I played Talespin on the NES years ago and it was, and likely is still, the HARDEST of the NES DA games. Best bet for this achievement would be Ducktales on Normal or Easy difficulty.

Happy hunting!