The big one - Gatekeeper

Gatekeeper for iOS would allow users to install apps directly from any website. The developer would still need to sign the apps with developer certificates that Apple controls and can revoke if they are used maliciously. The security model would still be intact, these apps would still live inside the sandbox so they couldn't hack other apps. They would however be able to use private APIs, which might have security implications in some cases.

Apple likes to control its Appstore which is a big source of revenue for them, so why in the world would they ever let go of the control of installed apps? This might seem counter-intuitive but I really think this would be great for Apple (and of-course everybody else).

If Apple allowed direct installs of apps, they don't have to be accountable for the software that gets installed on their customers' devices. Not only would Apple sign away a lot of problems, they would make their platforms more capable. No more backlash for cartoon-apps they reject, (no more rejection-backlash at all), a plethora of new types of apps that Apple don't like would suddenly pop up like mushrooms from the ground, making their platform more useful. Specific needs, like recording phone calls or downloading torrents (legal or otherwise), would suddenly be served by the iOS ecosystem. There would be a lot more iPads sold, since developers could then come up with their own business models to make developing pro-apps economically viable - as it seems to be on the mac (which has a fraction of the user-base of iOS). 

I don't even think it would hurt Apples 30% they get from Appstore sales, since the Appstore would still be the most convenient way to download apps - only those who can't (or aren't allowed to) make a living on the Appstore will publish outside it, and everybody will try to publish there first. Apps that generate income now, will still be there - there will just be an additional market. It also allows Apple to continue harassing its developers and be even harsher with stricter rules, since developers always have the choice to sell themselves. Freedom for both parties!

It would also create an incentive for Apple to improve their Appstore and accommodate the innovation that will happen outside it, in order to get as many developers in as possible (to earn those 30%). There will be other appstores with search that works and features they aren't able to figure out themselves (which they can then copy to improve their own store). They will benefit themselves tremendously by just entertaining the idea of giving up a slice of the 30%.

There are so many more fruits to pick

Not to be a total buzzkill but there are a few checkboxes Apple could tick in order to improve their platform tremendously, here is just a few on the top of my mind: 

• A setting to kill all animations - no fade in, just kill them. The devices would feel lightning fast without animations to play before you are allowed to interact.

• Animations MUST allow user interactions, its a simple flag! Just set it:

    UIViewAnimationOptionAllowUserInteraction
  

  While lesser developers might restrict their users, Apple should know better.

• Multi user iPad - this is a computer, it should act like one.

• Building an iPad homescreen you don't need to be ashamed about. Make use of that big screen.

• Allow me to set my own shortcuts on the control-center (the slide-up menu in the bottom of the device). The fact that Apple doesn't understand that everybody might not want the same functions here is just mind-boggling.
• Record phone-calls - it is actually a phone too!
• Allow for more/all types of apps, wouldn't it be cool to have an iPad web server? 
• Apple pen on all devices - why restrict functionality? 
• Two finger cursor movement/selection on all devices (3D touch is not good enough).
• A silent-mode that makes the phone 100% silent.

Apple will however do nothing of this. The reasoning seems to be that since functionality is imposing on aesthetics, which is holy, so functionality must be restricted.