A good game gives players reason to run over to their friends and family and say "you gotta see this!" or "I finally got that achievement!" They may not all be in the same room when it happens, so providing an outlet for people to brag from within the game itself in the middle of the action is the next best thing. Those shared posts may excite or at least interest other people, and they will be more inclined to play the game if they already have it, or get their own copy if they don't. "Refer a friend" buttons are good, too!
2. Support for carrying game progress to other devices or versions of the app
Using technology like iCloud to sync a player's progress among several devices and different apps of the same series can be a huge benefit. Often times, people will not upgrade from the free to the full version of a game because they can't take their level progress with them. Using the same sync store for multiple apps will allows sequels of your app, and even other games of yours to see what the player has played; and reward them with bonuses and unlocks.
3. A developer feedback button
It's no fun for a developer to discover new bugs and shortcomings of their game from comments in user reviews; especially ones that hurt their overall rating. It can happen if things just really bother the player, or if they just review everything they play. If you have a way to let them send you feedback in less than a few taps, you stand a chance of them deciding to send their comments to you directly. They might just give you a chance to fix whatever ails them and not write that bad review...so long as you deliver. Of course, they may also want to thank you and give you suggestions for some good new features!
4. Multiplayer or other community functionality
Building and maintaining a community of players is essential to the long-term success of a game. You can have a big successful product launch...and then have everyone stop playing after the first month. If you have players engaging each other, or developing content of their own to put into the game and share with others; then it adds whole new dimensions of things to do, and ways to have fun in the game. Competitions, tournaments, grinding, hunting, collaborations...and those are just some of the ones that developers have already come up with so far...
5. A Rate this app! button
My most recent game currently has over 13,000 downloads and less than 1% of players have rated it so far. How can prospective players trust it to be good if there's only 7 or 8 ratings in the market? For all they know, those all came from your friends and family. The game might not be inspiring enough to make people rate it...but maybe that's because it requires several taps to get back to the store's portal to do so. Letting the user get there in one tap might be the way to go. Just consider the risk involved: This is a good thing for good games, and a not-so-good thing for not-so-good games that might get not-so-good ratings.