Platforms: iOS, Android
You've got two choices at hand. Choice number one is to go on the app store and spend hours and hours looking for games, painstakingly waiting for them to download (especially if you've got slow internet), trying them out and then sighing in possible frustration when you realise this isn't really that fun. Choice number two is to download “Bored Button”, an app with a possibly infinite number of arcade games, for free.
The function of this app is extremely simple. You turn it on, it loads up a game from their server and you play it. If you don't like what you're playing, no need to fret – just press the nice, big red button and it immediately loads up another game for you to try out. If you STILL don't like it, rinse and repeat. If you like it, there's a save button which allows you to save all your favourite games.
The only drawback (apart from ads) is that some of the games here may not interest you and almost all the games tend to follow more or less the same pattern as the one you might have previous played.
On the plus side, if you have low expectations and don't play very taxing games on your phone, this app is all you need. Some games actually are pretty good.