- Edit -
The achievement unlocked for me when 478 full games completed was indicated. The counter is obviously glitchy. I'd say you should use it as a general guide.
- End of Edit -
The amount of full games you have completed is shown on the main menu, under the "Ranks & Stats" bit. Once it hits 500, the achievement should unlock. (It is sometimes slow to update, so restarting the app to get accurate stats is recommended)
Let's do some quick maths. Each game lasts for 150 seconds, with a 50 second break in between each game. That makes a "full game" 200 seconds long.
500x200=100000 seconds. Or, to put it another way, 27.7 hours. Ummm...good luck!
As people pointed out in the comments, you have to score for the game to count. You also cannot leave the game, so if you accidentally press the back button like I did countless times, it won't count.
One method suggested was to make a single snap at the start of the game, and let it run idle for the next 3 minutes. This will let you work towards the achievement while doing something else, however I recommend playing through the games as normal to get the 2,500,000 points required for the "Addict" achievement.