When people beat games easily or pass many levels without facing troubles they usually lose interest in the game. A game that is hard to the extent that people face big difficulties facing its first few levels is more likely to become addictive. Example: Flappy Bird.
Mini games and games that can be played in very short breaks are more likely to become addictive since people revert to them in the middle of other tasks whenever they feel like procrastinating.
When humans anticipate rewards their brains produce dopamine, a chemical that causes addiction. When the reward system in a game is implemented correctly people become more addicted to the game. Candy Crush is a very good example of a game that makes rewards emotional.
People get addicted to gambling because they can dream about any reward and still believe they can get it. When a game includes a variable reward instead of a fixed reward system, it's chances of becoming addictive become much higher. See the reasons why people gamble.
It was found that games that are simple to learn, not to beat, usually become more addictive since more people are likely to try them and stay hooked to them. See why simple products are more likely to succeed than complex ones.
Many of the addictive games don't have a predefined end. This prevents players from ever getting the feeling that they are done with the game. Subway Surfers is a good example of a game that doesn't have an end.
The human brain can be conditioned to experience pleasure when it gets certain rewards. Many games first get the attention of some players then condition their brains to enjoy it by manipulating their reward system.
Many games make people feel negative emotions then eliminate them quickly; thus creating a pain-pleasure cycle. Tetris is one of the games that has done this pretty well. It uses the frustration-relief cycle to keep people hooked.
Humans are social beings. Games that utilize social connections and helps people connect with each other are more likely to become popular and addictive.
The human mind usually believes that a scarce resource is more valuable. Some games utilize scarcity by giving players very limited resources that regenerate as the time passes. Clash of Clans is one good example of a game that does that well.
One of the addictive elements present in many games is the fact that things keep going on even when the player is away. This concept motivates players to come back and keeps them thinking about the game. Farmville is an example.
Games that compare scores of people to scores of their friends or others in their social circle can become much more addictive since they trigger the competitive spirit of competitive people.
Games that allow players to express themselves in their own way usually become more popular as they appeal to different classes of people. Minecraft is a perfect example of a game that utilizes that concept.
Most games provide some sort of a sense of progress to keep the player engaged even more. See examples of how a sense of progress motivates people.