"Minesweeper"
"Minesweeper" is free for iOS, although you can pay $2 for a premium version on iPhones. It's also free for Android, but there are ads.
"Snake"
There are lots of "Snake" games that emulate the classic Nokia game, but there's also a version called "Slither.io," where you can play against other "Snake" fanatics around the world.
It's free for iOS and Android. You can also play in a browser for free at slither.io.
"Galaga"
If you miss when game machines were huge cabinets and didn't fit in your pocket, you should download "Galaga," which is officially licensed.
It's free for iOS and Android, but you can pay money for ships and power-ups.
"Pac-Man"
The arcade classic is free for both iOS and Android.
"Chess"
The ancient game of skill is free for iPhones and Android. Get the version from Chess.com — it's free and lets you play with your friends or people around the world.
"Scrabble"
The officially licensed "Scrabble" app is free for iPhone and Android with in-app purchases.
There's also a more social version for iOS and Android — although it's not the official board — called "Words With Friends."
"Sonic the Hedgehog"
The classic Sega game is free-to-play, although a premium version is a $2 in-app purchase. Available for iOS and Android.
"Asteroids"
The classic space shooter is free with in-app purchases for iPhone and Android.